Solid Mensuration - Chapter 1

R I C H A R D T. E A R N H A R T

SOLID MENSURATION:

Chapter I

Introduction

Point, line, and plane are undefined terms in geometry. Using these undefined terms, other geometric figures are defined. Plane geometry is the study of

geometric figures that can be drawn on a two-dimensional surface called

plane. Figures that lie on a plane are called two-dimensional figures or simply plane figures. This chapter deals with different plane figures, and their

properties, relations, and measurement. The most common plane figures are the polygons.

Polygons

A polygon is a closed plane figure formed by line segments.

Parts of a Polygon

line segments that make up the polygon.

Adjacent sides are pairs of sides that share

a common endpoint.

2. The vertices of a polygon are the end points of each side of the polygon. Adjacent

vertices are endpoints of a side.

3. A diagonal of a polygon is a line segment joining two non-adjacent vertices of the polygon.

4. An interior angle is the angle formed by two adjacent sides inside the polygon.

5. An exterior angle is an angle that is adjacent to and supplementary to an interior angle of the polygon.

Side or Edge Vertex Diagonal Interior Angle Exterior Angle

A polygon may also be defined as a union of line segments such that:

i) Each vertex is a common end point of two adjacent line segments;

ii) no two adjacent line segments intersect except at an endpoint; and

Types of Polygons

1. Equiangular Polygon

A polygon is equiangular if all of its angles are congruent.

2. Equilateral Polygon

A polygon is equilateral if all of its sides are equal. 3. Regular polygon

Regular polygons are both equiangular and equilateral.

4. Irregular Polygon

A polygon that is neither equiangular nor equilateral is said to be an irregular polygon.

5. Convex Polygon

Every interior angle is less than 180°. If a line is drawn through the convex polygon, the line will intersect at most two sides.

6. Concave Polygon

A concave polygon has at least one interior angle that measures more than 180°. If a line is drawn through a concave polygon the line mat intersect more than two sides.

An example of a convex polygon

Naming Polygons

Polygons are named according to their number of sides. Generally, a polygon with n sides is called an n-gon. To form the name of polygons with 13 to 99 sides, begin with the prefix of the tens digit, followed by kai (the Greek word for and) and the prefix for the units digit.

Number of Sides Name of Polygon

n n-gon 3 triangle or trigon 4 quadlerateral or tetragon 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon or enneagon 10 decagon 11 undecagon or hendecagon 12 dodecagon 13 tridecagon or triskaidecagon 14 tetradecagon or tetrakaidecagon

Number of Sides Name of Polygon 16 hexadecagon or hexakaidecagon 17 heptadecagon or heptakaidecagon 18 octadecagon or octakaidecagon 19 enneadecagon or enneakaidecagon 20 isosagon 30 triacontagon 40 tetracontagon 50 pentacontagon 60 hexacontagon 70 heptacontagon 80 octacontagon 90 enneacontagon 100 hectogon or hecatontagon 1,000 chiliagon 10,000 myriagon 108 _megagon

Sides Prefix and Sides (Ones Digit) Suffix 20 icosi or icosa kai + 1 henagon 30 triaconta 2 digon 40 tetraconta 3 trigon 50 pentaconta 4 tetragon 60 hexaconta 5 pentagon 70 heptaconta 6 hexagon 80 octaconta 7 heptagon 90 enneaconta 8 octagon 9 enneagon

For numbers from 100 to 999, form the name of the polygon by starting with the prefix for the hundreds digit taken from the ones digit, affix the word hecta, then follow the rule on naming polygons with 3 to 99 sides. However, one may use the form n-gon, as in 24-gon for a poly24-gon with 24 sides, instead of using the above method.

Example 1

A 54-sided polygon is called a pentacontakaitetragon.

50 and 4

pentaconta kai tetragon

Example 2

A 532-sided polygon is called a pentahectatriacontakaidigon.

500 30 and 2

Similar Polygons

The ratio of two quantities is the quotient of one quantity divided by another quantity. Note, however, that the two quantities must be of the same kind. For example, the ratio of the measure of a side and an interior angle is meaningless because they are not quantities of the same kind. A proportion is an expression of equality between two ratios. That is, if two ratios a:b and c:d are equal, then the equation a/b=c/d is a proportion. Thus, you can say that a and b are proportional to c and d.

Two polygons are similar if their corresponding interior angles are congruent and their corresponding sides are proportional. Similar polygons have the same shape but differ in size.

Consider the similar polygons below. y_{1 y} 2 x_{1 x} 2 A₁ A₂

The following relations between the two polygons are obtained using the concept of ratio and proportion:

1. The ratio of any two corresponding sides of similar polygons are equal.

2 1 2 1

y

y

x

x

_

2. The ratio of the areas of similar polygons is the square of the ratio of any two corresponding sides. 2 2 1 2 1















x

x

A

A

3. The ratio of the perimeters of similar polygons is equal to the ratio of any of any two corresponding sides.

2 1 2 1

x

x

P

P

_

Properties of a Regular Polygon

A regular polygon of n sides can be subdivided into n congruent isosceles triangles, whose base is a side of the polygon. The common vertex of these triangles is the center of the polygon.

s a

θ/2 Perimeter

To find a perimeter of a polygon, add the lengths of the sides of the polygon. Since regular polygons are equilateral, the formula in finding the perimiter of a regular polygon is

Central Angle

The angle that is opposite a side of a regular polygon is a central angle θ of the polygon. It is the angle formed by two lines drawn from the center of the polygon to two adjacent vertices. Regular polygons are equiangular. Thus, the measure of each angle is given by

n

360





The altitude of the isosceles triangles that can be formed from a regular polygon is the apothem of the regular polygon. The apothem bisects the central angle and its opposite side. Thus, we can compute for the apothem as follows:

a s n 2 2 360 tan 2 tan     2 s a 2  Solving for a,        n s a _ 180 tan 2

Interior Angle

In each isosceles triangle, the measure of the base angles can be denoted by Φ, and each interior angle of the regular polygon by 2Φ. Thus, the measure of each interior angle is solved as follows:









Sum of Interior Angles

Since the number of sides equals the number of interior angles, then the sum of interior angles is n times the measure of the interior angle. Hence,



2



180

.

.

I

A



n



S

From any given vertex of a regular polygon, a diagonal is drawn from the

vertex to a non-adjacent vertex. This means that you can construct a diagonal from each vertex of a polygon with n sides in n – 3 ways. Since there are n vertices and each diagonal has two end points, you can do this in only 𝑛₂ ways. Thus, the total number of distinct diagonals of a regular polygon is

𝐷 = 𝑛

Area

Area is the amount of two dimensional space that a plane figure occupies. To get the area of a regular polygon, multiply the area of the isosceles triangle by the number of triangles formed, or 𝐴 = 𝑠₂ (𝑎)(𝑛).

Note that this is the same as one half of the product of its perimeter and its apothem. In general, the formula of a regular polygon is given by

𝐴 = 1₂ Pa

The formula for area of a regular polygon can be expressed in terms of its number of sides and the measure of one side as follows:

𝐴 = 1 2𝑃𝑎 = 1₂ 𝑛𝑠 𝑠 2𝑡𝑎𝑛 180°_𝑛 𝐴 = 𝑛𝑠 2 180°

Example 3

Find the area of a regular nonagon whose sides measure 3 units. Determine the number of distinct diagonals that can be drawn from each vertex and the sum of its interior angles.

Solution:

A nonagon is a 9-sided polygon. Thus, n = 9. Given s = 3, solve as follows: Area of the polygon:

𝐴 = 𝑛𝑠 2 4𝑡𝑎𝑛 180°_𝑛 = 9(3) 2 4𝑡𝑎𝑛 180°₉ =55.64 square units Number of diagonals: 𝐷 = 𝑛 2 𝑛 − 3 = 9 2(9 − 3) = 27

Sum of interior angles:

𝑆. 𝐼. 𝐴. = 180° 𝑛 − 2 = 180°(9 − 2)

Exercises

1. Use the diagram below to answer questions (a) to (d)

A

a) Is the polygon convex or concave?

b) How many diagonals can be drawn from vertex A? c) How many sides does the polygon have?

2. Use the diagram below to answer questions (a) to (d)

A

a) Is the polygon convex or concave?

b) How many diagonals can be drawn from vertex A? c) How many sides does the polygon have?

3. Find the measure of an interior angle of a regular tridecagon.

4. That is the measure of an interior angle of a regular pentacontakaitrigon? 5. Find the sum of the interior angles of a regular trcontakaitetragon.

6. What is the sum of the interior angle of a regular icosagon?

7. Name each polygon with the given number of sides. Also, find the number of diagonal of each polygon.

a) 24 b) 181 c) 47 d) 653

8. Name each polygon with the given number of sides. a) 39

b) 127 c) 821 d) 86

9. How many sides does each polygon have? a) Icosikaihenagon

b) Enneacontakaidigon

10. How many sides does each polygon have? How many distinct diagonals can be drawn from a vertex of each polygon?

a) Trihectatriacontakaitrigon b) Pentacontakaioctagon

c) Heptacontakaiheptagon

11. The number of diagonals of a regular polygon is 35. Find the area of the polygon if its apothem measures 10 centimiters

12. The number of diagonals a regular polygon is 65. Find the perimiter of the polygon if its apothem measures 8 inches.

13. The Sum of the interior angles of a regular polygon is 1,260˚. Find the area of the polygon if its perimeter is 45 centimeters.

14. The measure of an interior angle of a regular polygon is 144˚. Find the apothem if one side of the polygon measures 5 units.

15. Find the number of sides of each of the two polygons if the total number of sides of the polygons is 13, and the sum of the number of diagonals of the polygons is 25.

16. Find the number of sides of each of the two polygons if the total number of sides of the polygons is 15, and the sum of the number of diagonals of the polygon is 36.

17. What is the name of a regular polygon that has 90 diagonals? 18. What is the name of a regular polygon that has 135 diagonals?

19. Find the number of diagonals of a regular polygon whose interior angle measures 144˚

20.Find the sum of the interior angles and the number of diagonals of a regular polygon whose central angle measures 6˚.

21. The ratio of areas between two similar triangles is 1:4. If one side of the smaller triangle is 2 units, find the measure of the corresponding side of the other triangle.

22.One side of a polygon measures 10 units. If the measure of the corresponding side of a similar polygon is 6 units, find the ratio of their areas. What is the area of the larger polygon if the area of the smaller polygon is 12 square units? 23. A regular hexagon A has the midpoints of its edges joined to form a smaller

hexagon B. This process is repeated by joining the midpoints of the edges of hexagon B to get a third hexagon C. What is the ration of the area of hexagon C to the area of hexagon A?

24.What is the ratio of the area of hexagon B to the area of hexagon A in number 23?

25. If ABCDE is a regular pentagon and diagonals EB and AC intersect at O, then what is the degree measure of angle EOC?

Triangles

The most fundamental subset of polygons is the set of triangles. Although triangles are polygons with the least number of sides, these polygons are widely used in the field of mathematics and engineering. In this section, some important formulas which are used extensively in solving geometric problems will be introduced.

Classification of Triangles According to Sides

1. Equilateral – a triangle with three congruent sides and three congruent

angles. Each angle measures 60˚.

2. Isosceles – a triangle with two congruent sides and two congruent angles. 3. Scalene – a triangle with no congruent sides and no congruent angles.

60° 60° 60° Equilateral Isosceles θ θ Scalene

Classifications of Triangles According to Angles

1. Right – a triangle with a right angle (90˚ angle). 2. Oblique – a triangle with no right angle.

a) Acute – a triangle with three acute angles (less than 90˚)

b) Equiangular – a triangle with three congruent angles. Each angle

measures 60˚.

c) Obtuse – a triangle with one obtuse angle (more than 90˚ but less than

180˚

Congruent Triangles

The word congruent is derived from the Latin word congruere which means agree. Two triangles are congruent when they have the same shape and size.

Congruent triangles can be made to coincide part by part. Corresponding parts of congruent triangles are congruent. The symbol for congruence is ≅

Similar Triangles

Two triangles are similar if their corresponding sides are proportional. Similar triangles have the same shape but differ in size. Look at the similar triangles below.

a₁ a2

b₁

b₂

c₁ c2

Since the two triangles are similar, then the relations that exist between two similar polygons also hold. Thus, it follows that:

a) 𝑎1 𝑎₂ = 𝑏₁ 𝑏₂ = 𝑐₁ 𝑐₁ b) 𝐴1 𝐴₂ = 𝑎₁ 𝑎₂ 2 = 𝑏1 𝑏₂ 2 = 𝑐1 𝑐₂ 2 c) 𝑃1 𝑃₂ = 𝑎₁ 𝑎₂ = 𝑏₁ 𝑏₂ = 𝑐₁ 𝑐₁

Parts of a Triangle

A triangle has three possible bases and three possible vertices. Any of the three sides of a triangle may be considered as the base of the triangle. The angle opposite the base is called vertex angle. The two angles adjacent to the base are called base angles.

A line segment drawn from a vertex perpendicular to the opposite side is called

altitude. The point of intersection of the altitudes of a triangle is called orthocenter. A median of a triangle is the line segment connecting the

midpoint of a side and the opposite vertex. The centroid is the point of intersection of the medians of a triangle. An angle bisector divides an angle of the triangle into two congruent angles and has endpoints on a vertex and the opposite side. The point of intersection of the angle bisectors of a triangle is called incenter. Orthocenter Incenter Centroid A B/2 A/2 A/2 C/2

A perpendicular bisector of a side of a triangle divides the side into two congruent segments and is perpendicular to the side. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. The Euler line is the line which contains the orthocenter, centroid, and circumcenter of a triangle. The centroid is located between the orthocenter and the circumcenter.

However, in an equilateral triangle, the centroid, circumcenter, incircle, and orthocenter are coincident.

Circumcenter Perpendicular Bisectors Orthocenter Circumcenter Centroid Euler Line

Properties of Triangle Centers

1. Orthocenter – The orthocenter is not always in the interior of the triangle. In

an obtuse triangle, the two sides of the obtuse angle and the corresponding

altitudes are extended to meet at a point outside the triangle. In a right triangle, the orthocenter is on a vertex of the triangle.

2. Centroid – The centroid is known as the center of mass of the triangle. Unlike

the orthocenter, the centroid is always inside the triangle and for right, isosceles and equilateral triangles, the centroid is located one-third of the altitude from the base.

3. Incenter – The incenter is the center of the largest circle that can be inscribed

in the triangle.

4. Circumcenter – The circumcenter is the center of the circle circumscribing a

triangle. It is not always inside the triangle. The vertices of the triangle lie on the circle and are equidistant from the circumcenter.

Altitude, Median, and Angle Bisector Formulas

Consider an arbitrary triangle with sides a, b, and c, and angles A, B, and C/ Let h_c, m_c and I_c be the lengths of the altitude, median, and angle bisector from vertex C, respectively. Then, 𝑕_𝑐 = 2 𝑠(𝑠;𝑎)(𝑠;𝑏)(𝑠;𝑐)_𝑐 , C A B b a c h_c Altitude:

Where s is the semi-perimeter of the triangle and 𝑠 = 𝑎:𝑏:𝑐

2 , C A B b a c m_c Median: Angle Bisector: 𝑚_𝑐 = 1 2 2𝑎2 + 2𝑏2 − 𝑐2 𝐼_𝑐 = 𝑎𝑏 𝑎 + 𝑏 2 − 𝑐2 𝑎 + 𝑏 C b a m_c

Facts About Triangles

1. The sum of the lengths of any two sides of a triangle is always greater than the third side. The difference between the lengths of any two sides is always less the third side of a triangle.

2. The sum of the measures of the interior angles of a triangle is 180˚. 3. Two equiangular triangles are similar.

4. Two triangles are similar if their corresponding sides are parallel. Two triangles are similar if their corresponding sides are perpendicular.

5. In any right triangle, the longest side opposite the right angle is called hypotenuse.

6. If any two sides of a right triangle are given, the third side can be obtained by the Pythagorean Theorem c2_=a2_+b2_.

7. Two triangles are equal if the measures of the two sides and the included angle of one triangle are equal to the measures of the two sides and the included angle of the other triangle.

8. The line segment which joins the midpoints of two sides of a triangle is parallel to the third side and equal to one-half the length of the third side.

9. In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

10. The altitude h to the hypotenuse c of a right triangle divides the triangle into two similar triangles. Each of the triangles formed by this altitude is similar to the original triangle.

11. Each leg of a right triangle is the geometric mean between the hypotenuse and the projection of the leg on the hypotenuse.

h x c - x a p c 𝑎 = 𝑐𝑝

Formulas for the Area of the Triangle

In general, the area of any triangle is one-half the product of its base and its altitude.

𝐴 = 1 2𝑏𝑕

To solve for the area of a triangle given the measures of two sides and an included angle, use the SAS formula.

SAS (Side-Angle-Side) Formula

𝐴 = 1

2𝑎𝑏 𝑠𝑖𝑛𝜃

a b 𝜃

The area of a triangle is one-half the product of any two sides and the sine of their included angle.

When the measure of the three sides of a triangle are given, the area of the triangle is determined by Heron’s Formula.

Heron’s Formula or SSS (Three Sides) Formula:

𝐴 = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐), C A B b a c

Example 4

The measures of the three sides of a triangle are AB = 30 in., AC = 50 in., and BC = 60in.. From a point D on side AB, a line DE is drawn through a point E on side AC such that angle AED is equal to angle ABC. If the perimeter of the triangle ADE is equal to 56 in., find the sum of the lengths of line segments BD and CE.

Solution:

Draw the figure and label the parts with the given measures.

B A E D C 60 𝜃 30 50

The perimeter of triangle ABC is P = 30 + 60 + 50 = 140 in. Notice that triangle ABC and triangle ADE are similar triangles since ABC ≅ AED and BAC ≅

DAE. Thus, the ratio of their perimeters is equal to the ratio of any of their

corresponding sides. 56 140 = 𝐴𝐷 50 → 𝐴𝐷 = 20 56 140 = 𝐴𝐸 30 → 𝐴𝐸 = 12 𝐵𝐷 = 30 − 𝐴𝐷 𝐸𝐶 = 50 − 𝐴𝐸 = 30 − 20 = 50 − 12 = 10 = 38 Hence, BD + EC = 10 + 38 = 48 in.

Example 5

Derive formulas for the height and area of an equilateral triangle with side s.

Solution:

In an equilateral triangle, the altitude divides the triangle into two congruent right triangles. Thus, by the Pythagorean Theorem,

60˚ s h 𝑠 2 𝑕 = 𝑠2 ₋ 𝑠 2 2

=

𝑠

Since each interior angle measures 60°, use the SAS formula to find the area of the triangle.

𝐴 = 1 2𝑠2 sin 60° = 1 2𝑠2 3 2 = 3𝑠2

Example 6

If one side of a triangle is 20 units and the perimeter is 72 units, what is the maximum area that the triangle can have?

Solution:

Imagine the side of the length 20 units as the base of the triangle. Thus, the sum of the lengths of the other two sides is P – 20 = 52 units. Since the area of the triangle is maximum when the height is also maximum, the triangle is isosceles and the two sides measure 26 units each. By Pythagorean Theorem,

𝑕 = 262 _{− 10}2 _{= 24}

Hence, the area is 𝐴 = 1 2𝑏𝑕 = 1₂ 20 24 = 240 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠 26 h 10 26 10

Example 7

Derive the formula for the median of triangle ABC drawn from vertex C to side AB using the Cosine Law.

Solution:

Draw and label the triangle. by the Cosine Law, you get:

𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos 𝐵 C A B b a 𝑐 2 𝑐 2 Adding the two equations yields the identity

𝑐 = 𝑎 cos 𝐵 + 𝑏 cos 𝐴 (1)

Apply the Cosine Law to each of the triangles formed by the median to get: 𝑚2 = 𝑏2 + 𝑐 2 2 − 2𝑏 𝑐 2 cos 𝐴 cos 𝐴 = 𝑏 2_:𝑐2 4;𝑚2 𝑏𝑐 (2) 𝑚2 = 𝑎2 + ₂𝑐 2 − 2𝑎 𝑐₂ cos 𝐵 𝑎2:𝑐2₄;𝑚2

Substituting equations 2 and 3 in equation 1 yields 𝑐 = 𝑎 𝑎 2_:𝑐2 4;𝑚2 𝑎𝑐 + 𝑏 𝑏2:𝑐2₄;𝑚2 𝑏𝑐

Solving for m, you get

EXERCISES

1. Is it possible to form a triangle with sides 20, 30, and 50 units? Explain.

2. Is it possible to form a triangle with sides 2, 4, and 8 units? Justify your answer. 3. Find the altitude and the area of an equilateral triangle whose side is 8 cm long. 4. One side of an isosceles triangle whose perimeter is 42 units measures 10 units.

Find the area of the triangles

5. Find the area of an equilateral if its altitude is 5 cm.

6. The ratio of the base of an isosceles triangle to its altitude is 3:4. Find the measures of the angles of the triangle.

7. The base of an isosceles triangle and the altitude drawn from one of the congruent sides are equal to 18 cm and 15 cm, respectively. Find the length of the sides of the triangle.

8. Two altitudes of an isosceles triangle are equal to 20 cm and 30 cm. Determine the possible measures of the base angles of the triangle.

9. In a right triangle, the bisector of the right angle divides the hypotenuse in the ratio of 2 is to 5. Determine the measures of the acute angles of the triangle. 10. The area of a triangle is equal to 48 cm2 _{and two of its sides measure 12 cm and}

9 cm, respectively. Find the possible measures of the included angles of the given sides.

12. Find the area of a triangle if its two sides measure 6 in. and 9 in., and the bisector of the angle between the sides is 4 3 in.

13. In an acute triangle ABC, an altitude AD is drawn. Find the area of triangle ABC if AB = 15 in., AC = 18 in., and BD = 10 in.

14. In a right triangle, a line perpendicular to the hypotenuse drawn from the midpoint of one of the sides divides the hypotenuse into segments which are 10 cm and 6 cm long. Find the lengths of the two sides of the triangle.

15. Given triangle ABC whose sides are AB = 15 in., AC = 25 in., and BC = 30 in. From a point D on side AB, a line DE is drawn to a point E on side AC such that angle ADE is equal to angle ABC. If the perimeter of triangle ADE is 28 in., find the lengths of line segments BD and CE.

16. Suppose that AD, BC, AC and BD are line segments with line AD parallel to line BC as shown in the figure on the right. If AD = 3 units, BC = 1 unit, and the distance from

AD to BC is 5 units, find the altitude of the smaller

triangle.

A 3 D

B ₁

5

17. What is the sum of the areas of the two triangles formed in number 16? 2 A C E B D F

18. If ∆𝐴𝐵𝐶 is equilateral,

=

,

=

,

and

=

3

. Find the ratio of the area of

∆𝐴𝐵𝐶 to the shaded area.

19. In triangle ABC, E is the midpoint of AC and D is the midpoint of CB. If DF is parallel to BE, find the length of side AB.

A B C _D E F 3 4 5

20. The measure of the base of an isosceles triangle is 24 cm, and one of its sides is 20 cm long. Find the distance between the centroid and the vertex opposite the base.

21. The two sides of a triangle are 17 cm and 28 cm long, and the length of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to the longest side.

QUADRILATERALS

A quadrilateral, also known as tetragon or quadrangle, is a general term for a four-sided polygon. There are six types of quadrilaterals. They are square, parallelogram, rectangle, rhombus, trapezoid, and trapezium. Each type of quadrilateral has unique properties that make it distinct from other types. A square is the most unique quadrilateral because it possess all those unique properties.

The common parts of a quadrilateral are described as follows:

1. Side – A side is a line segment which joins any two adjacent vertices.

2. Interior angle – An interior angle is the angle formed between two adjacent

sides.

3. Height or Altitude – It is the distance between two parallel sides of a

quadrilateral.

4. Base – This is the side that is perpendicular to the altitude.

Classification of Quadrilaterals

The classification of quadrilaterals is based on the number of pairs of its parallel sides as shown in the figure below.

Quadrilateral Parallelogram Rectangle Square Rhombus Trapezoid Trapezium Classifications of Quadrilaterals

• Parallelogram has two pairs of parallel sides.

• Trapezoid has only one pair of parallel sides.

• Trapezium does not have any pair of parallel

sides.

• Rectangle, rhombus, and square are special

General Formulas for the Area of Quadrilaterals

Consider the quadrilateral below.

A B C D a b c d e₁ e𝜃 ₂

There are several useful formulas for the area of a planar convex quadrilateral in terms of sides a, b, c, and d, and diagonal lengths e₁ and e₂. Among them are the

following: 𝐴 = 1

2𝑒1𝑒2 sin 𝜃 ,

Formula 1: where 𝜃 is the angle formed between e₁ and e₂.

Formula 2: 𝐴 = 1₄ 𝑎2 + 𝑐2 − 𝑏2 − 𝑑2 tan 𝜃 , where the four sides are labeled _{such that a2 + c2 > b2 + d2.} Formula 3: 𝐴 = 𝑠 − 𝑎 𝑠 − 𝑏 𝑠 − 𝑐 𝑠 − 𝑑 − 𝑎𝑏𝑐𝑑 cos2 1

2 𝐴 + 𝐶 ,

where s is the semi-perimeter and angles A and C are any two opposite angles of the quadrilateral.

Note that in Formulas 1 and 2, sin 𝜃 = sin(180° − 𝜃) and tan 𝜃 = tan(180° − 𝜃 . Thus, you can choose the other angle formed by the two diagonals without affecting

PARALLELOGRAM

A parallelogram is a quadrilateral in which the opposite sides are parallel. The figure below illustrates an example of a parallelogram.

A B

C D

b (base)

h (height)

Parallelograms have the following important properties: 1. Opposite sides are equal.

2. Opposite interior angles are congruent (e.g., 𝐴 ≅ 𝐶).

3. Adjacent angles are supplementary (e.g., 𝐴 + 𝐷 = 180° ).

4. A diagonal divides the parallelogram into two congruent triangles (e.g., ∆𝐷𝐴𝐵 ≅ ∆𝐷𝐶𝐵).

Diagonals of a Parallelogram

If sides a and b, and the angle 𝜃 are given, then by the Cosine Law, the diagonal may be obtained by the equation:

𝑑2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝜃 A B C D h a b d 𝜃 a h 𝜃

If any two parts of the parallelogram are given, the relationship among a, h, and 𝜃 may be obtained from the right triangle. Using the other angle 180° − 𝜃, the second diagonal may be obtained by the same formula.

Perimeter of a Parallelogram

Opposite sides of a parallelogram are equal. Thus, its perimeter is given by 𝑃 = 2𝑎 + 2𝑏

Area of a Parallelogram

The area of a parallelogram can be obtained by any of the following formulas:

Formula 1:

where b is the length of the base, and h is the height. 𝐴 = 𝑏𝑕,

Formula 2: 𝐴 = 𝑎𝑏 sin 𝜃

where a and b are the lengths of the sides of the parallelogram and θ is any interior angle.

In problems involving area of a parallelogram, one will have to find the height h or the angle θ using the right triangle containing the parts a, h, and θ. Since a diagonal divides the parallelogram into two congruent triangles, the area of the parallelogram is twice the area of one of the two triangles. Thus, if two sides a and b, and an angle θ are given, you can obtain the area using SAS formula. The area of the parallelogram is determined by 𝐴 = 2 1₂𝑎𝑏 sin 𝜃 = 𝑎𝑏 sin 𝜃. Note theta the angle is any interior angle since 𝑠𝑖𝑛𝜃 = 𝑠𝑖𝑛 180° − 𝜃

Kinds of Parallelogram

The next three quadrilaterals that will be discussed-rectangles, rhombuses, and squares-are all special types of parallelograms. You can classify each shape

depending on the congruent sides and angles. Given a shape, you can work

backwards to find out its sides or angles. Coordinate geometry is an effective way to measure the angles and the sides

RECTANGLE

A rectangle is essentially a parallelogram in which the interior angles are all right angles. Since a rectangle is a parallelogram, all of the properties of a

parallelogram also hold for a rectangle. In addition to these properties, the

diagonals of a rectangle are equal. However, the sides are not necessarily all equal.

A B

C D

h

Diagonals of a Rectangle

A diagonal of a rectangle cuts the rectangle into two congruent right triangles. In the figure on page 26, the diagonal AC divides the rectangle ABCD into congruent right triangles ADC and ABC. Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, one can always compute for the third side with the use of the Pythagorean Theorem, if any two of these parts are given. Thus, the diagonal d=AC may be determined using the equation

𝑑 = 𝑏2 _{+ 𝑕}2

Perimeter of a Rectangle

The perimeter is the sum of the four sides. Thus, 𝑃 = 2𝑏 + 2𝑕.

Area of a Rectangle

If b is the length of the base and h is the height, then the formula for the area of a rectangle is

SQUARE

A square is a special type of a rectangle in which all the sides are equal. Since all sides and interior angles are equal, a square is classified as a regular polygon of four sides.

a d

a

Diagonal of a Square

The steps in finding the diagonal of a square is similar to the steps used in a rectangle. Thus, 𝑑 = 𝑎2 _{+ 𝑎}2 _{= 𝑎 2, where a is the length of one side of the}

square.

where a is the length of one side of the square. 𝑑 = 𝑎 2

Note that, if the length of the diagonal is given, one can always compute for the length of the sides of the square using the same formula.

Perimeter of a Square

Since all the sides of a square are equal, it is also possible to provide a simple formula for the perimeter of the square. Thus, the simplified form of the perimeter is

𝑃 = 4𝑎

Area of a square

The formula for the area of a square is given by 𝐴 = 𝑎2.

RHOMBUS

A rhombus is a parallelogram in which all sides are equal.

b d₁ 𝜃

d₂

h

A rhombus may also be defined as an

equilateral parallelogram. The terms

“rhomb” and “diamond” are sometimes used instead of rhombus. A rhombus with an interior angle of 45ᴼ is sometimes called a lozenge.

The Diagonal of a Rhombus

Just like the square, the diagonals of a rhombus are perpendicular bisectors. Thus, the angles formed by the diagonals measure 90ᴼ and the length of each side of the rhombus is given by 𝑏 = 𝑑1

2 2 + 𝑑2 2 2 b 𝑑₁ 2 𝜃 2 h 𝜃 𝜃 2 𝑑₂ 2

Also, the diagonals of the rhombus are angle bisectors of the vertices. By the Cosine Law, the diagonals may be obtained in a similar manner like that of a parallelogram. Thus,

𝑑₁2 = 2𝑏2 1 − 𝑐𝑜𝑠𝜃 and 𝑑₂2 = 2𝑏2 1 + 𝑐𝑜𝑠𝜃

One can also verify that the angle opposite the shorter diagonal d₁, may be obtained by the formula

𝜃 = 2𝑡𝑎𝑛;1 𝑑1

𝑑₂ .

where d₂ is the longer diagonal and θ is the angle opposite the shorter diagonal.

The Perimeter of a Rhombus

If b is the measure of one side of a rhombus, then the perimeter is given by 𝑃 = 4𝑏.

Area of a Rhombus

The area of a rhombus may be determined by any of the following ways: The area is one-half the product of its two diagonals.

𝐴 = 1

2𝑑1𝑑2

Note that this expression follows from Formula 1 for the area of quadrilateral, where θ=90ᴼ

Since a rhombus is a parallelogram, the area is also the product of the base times the height.

𝐴 = 𝑏𝑕

The area is twice the area of one of the two congruent triangles formed by one of its diagonals. This is the same method used in finding the area of a

parallelogram.

TRAPEZOID

A trapezoid is a quadrilateral with one pair of parallel sides.

a b 𝜃 𝜃 h 𝒃 − 𝒂 𝟐

In the trapezoid shown above, the parallel sides a and b are called bases and h is the height or the perpendicular distance between the two bases. If the non-parallel sides are congruent, the trapezoid is called an isosceles trapezoid. The base angles of an isosceles trapezoid are also congruent. One can observe that the relationship among the sides, height, and base angles of an isosceles trapezoid may be obtained from the right triangle formed by constructing a line from one vertex perpendicular to the opposite side (lower base).

A trapezoid which contains two right angles is called a right trapezoid. The trapezoid on the right is an example of a right trapezoid.

a

b-a

h

b

Area of a Trapezoid

The area of a trapezoid is equal to the product of the mean of the bases and the height. In symbols, the area is given by the formula

𝐴 = 1₂ 𝑎 + 𝑏 h.

The median of a trapezoid is the line segment parallel to and midway between the bases of the trapezoid. Thus, 𝑚 = 𝑎:𝑏₂ and A=mh.

TRAPEZIUM

A trapezium is a quadrilateral with no parallel sides. In finding the area of a trapezium, you may use any of the three formulas for the area of a quadrilateral.

Example 8

Find the area and perimeter of a square whose diagonal is 15 units long.

a

a 15

First find the length of a side of the square using the formula 𝑑 = 𝑎 2. Thus, the

measure of the side of the square is 𝑎 = 15 2₂ units.

Therefore, the area is A=112.5 square units and the perimeter is 𝑃 = 30 2, or 42.43 units.

Example 9

The side of a square is x meters. The midpoints of its sides are joined to form another square whose area is 16 m2_{. Find the value of x and the area of the} portion of the bigger square that is outside the smaller square.

Solution:

Let y be the measure of one side of the inscribed square. Since you know that the area of the inscribed square is 𝑦2, the value of y is 4. In triangle ABC,

42 = (𝑥 2)2 + ( 𝑥 2)2 Which yields x = 4 2 𝑚. A B C 𝑥 2 𝑥 2

The difference between the areas of the two squares is the area calculated as follows:

Required Area = 32 – 16 = 16 𝑚2

You can actually compute for the area by symmetry on the two figures knowing that the area of the bigger square is twice the area of the smaller one which

Example 10

If ABCD is a rhombus, AC=4, and ADC is an equilateral triangle, what is the area of the rhombus?

Solution:

If ADC is an equilateral triangle, the then length of a side of the rhombus is 4, and angle ADC is 60°.

Thus, the area of the rhombus is 𝐴 = 2 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐷𝐶 = 2 1₂ 4 4 𝑠𝑖𝑛60° = 8 3 ≈ 13.86 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠 A _B C D 4

Example 11

Find the diagonal of the rectangle inscribed in the isosceles right triangle shown in the figure if the upper two vertices of the rectangle lie at the midpoints of the two legs of the triangle.

A B C D E F G 12m Solution: In triangle ABC, 𝐴𝐵 12 = 𝑠𝑖𝑛45° AB= 8.49m

Since E is the midpoint of BC,

BE = BD = EC = 𝐴𝐵₂ = 4.24m Hence, AD = BD = 4.24m.

DE = 1₂𝐴𝐶 = 1₂ 12 = 6𝑚 𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 In triangle EGC, _4.24𝐸𝐺 = 𝑠𝑖𝑛45°, which gives

EG = 3m (height of rectangle)

Hence, the diagonal of the rectangle is

Example 12

Find the area and the perimeter of the right trapezoid shown in the figure. 8

11

60°

Solution:

To find the area, draw the height of the trapezoid such that a right triangle is formed as shown in the figure below. The length of the base and the height of this triangle are 3 and h, respectively, where:

h = 3 tan 60° = 5.2 units

and z = _{cos 60°}3 = 6 𝑢𝑛𝑖𝑡𝑠

Thus, the area and the perimeter of the trapezoid are:

𝐴 = 1₂ 𝑎 + 𝑏 𝑕 = 1₂ 8 + 11 5.2 = 49.4 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠

P= sum of all sides

P= 8 + 11 + 5.2 + 6 = 30.2 units 8 3 60° 8 h h z

Example 13

A vacant lot has the shape of a trapezium with sides 8m, 12m, 18m, and 20m. If the sum of the opposite angles is 230°, find the area of the lot.

8 12

20 18

𝜃₁ 𝜃2

Solution:

Referring to the figure to the right, compute: 𝜃1:𝜃₂ 2 = 230°₂ = 115°

The semi-perimeter is

𝑠 = 𝑎:𝑏:𝑐:𝑑₂ = 8:12:18:20₂ = 29.

Therefore, the area of the trapezium is

𝐴 = 𝑠 − 𝑎 𝑠 − 𝑏 𝑠 − 𝑐 𝑠 − 𝑑 − 𝑎𝑑𝑐𝑏𝑐𝑜𝑠2₍𝜃1:𝜃2

2 )

= 29 − 8 29 − 12 29 − 18 29 − 20 − 8(12)(18)(20)𝑐𝑜𝑠2_[115°]

Example 14

If the sides of the parallelogram and an included angle are 8m, 12m, and 120°, respectively. Find the length of the shorter diagonal and the area of the

parallelogram.

Solution:

In the figure shown on the right, 𝜃 = 180° − 120° = 60°. By Cosine law, 𝑑2 = 82 + 122 − 2 8 12 𝑐𝑜𝑠60° 𝑑 = 4 7m In triangle CDE, ℎ₈ = 𝑠𝑖𝑛60° 𝑜𝑟 𝑕 = 4 3 ≈ 6.93𝑚

Therefore, the area of the parallelogram is 𝐴 = 𝑏𝑕 = 12 4 3 = 48 3 ≈ 83.14𝑚2 Alternative Solution: 𝐴 = 2 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝐴𝐵𝐷 𝐴 = 2 1₂ 8 12 𝑠𝑖𝑛60° = 48 3𝑚2 A B C D E d 120° 8 𝜃 𝜃 h

Example 15

Verify the area of the parallelogram in example 14 using the three general formulas for the area of quadrilaterals.

Solution:

Referring to the previous example, obtain the following results: 𝑑₁ = 𝐵𝐷 = 4 7

𝑑₂ = 𝐴𝐶 = 82_{+ 12}2_{− 2 8 12 𝑐𝑜𝑠120° = 4 19} By Cosine Law,

82 = (4 7₂ )2+ (4 19₂ )2− 2 4 7₂ 4 19₂ 𝑐𝑜𝑠∅ ∅ = 64.31°

where 𝑑₁and 𝑑₂ are the two diagonals, and the acute angle ∅ is the included angle between these two diagonals.

Formula 1: The area of the parallelogram is

𝐴 = 1₂𝑑₁𝑑₂𝑠𝑖𝑛∅ =1₂ 4 7 4 19 𝑠𝑖𝑛64.31° = 83.14𝑚2

Formula 2: With a=c=12 and b=d=8, then

𝐴 = 1₄ 𝑎2_{+ 𝑐}2_{− 𝑏}2 _{− 𝑑}2 _{|𝑡𝑎𝑛∅|} = 1

4 12

2_{+ 12}2_{− 8}2_{− 8}2 _|tan64.31°| = 83.14𝑚2

Formula 3: With semi-perimeter s=20, obtain:

𝐴 = (20 − 8)2_{(20 − 12)}2 _{− (8}2₎₍₁₂2_{)𝑐𝑜𝑠}2_[1

2 60° + 60° ] = 83.14𝑚2

Example 16

A trapezoid has an area of 144𝑚2 and altitude of 4m. Its two bases have a ratio of 4:5. What are the lengths of the bases in meters? What is the perimeter of the trapezoid?

Solution:

The area of the trapezoid is: 144 = 1₂ 𝑎 + 𝑏 4 (E1) The sides ratio is: 𝑎_𝑏 = 4₅

𝑏 = 5₄𝑎(E2)

Substituting (E2) into (E1), 𝑎 = 32𝑚; 𝑎𝑛𝑑 𝑓𝑟𝑜𝑚 𝐸2 , 𝑏 = 40.

Now, to determine the perimeter, you only need to know the length of c, In triangle QRS, 𝑥 = 𝑏;𝑎₂ = 4

By Pythagorean Theorem, 𝑐 = 𝑕2 _{+ 𝑥}2

= 42 _{+ 4}2

= 4 2

Thus, the perimeter is

𝑃 = 40 + 32 + 2 4 2 = 83.31𝑚 Q a S c _c x h=4 R

Example 17

The area of a rhombus is 143𝑚2. If the longer diagonal is 26m, find the angle opposite the shorter diagonal and the length of one side of the rhombus.

Solution:

In the figure shown, you can compute for the length of the shorter diagonal based on the area of the rhombus. Thus, solve as follows:

𝐴 = 1₂𝑑₁𝑑₂ 143 = 1

2𝑑1 26

𝑑₁ = 11𝑚

Since the diagonal of a rhombus are perpendicular bisectors, the four triangles formed by these diagonals are congruent right triangles.

Thus, in one of these triangles, you can get the length of side x and the interior angle 𝜃 as show below. tan 𝜃 2 = 𝑑₁ 𝑑₂ 𝜃 = 2𝑡𝑎𝑛;1 𝑑1 𝑑₂ = 45.86°

The length of side x may be computed using the Pythagorean Theorem. 𝑥 = (𝑑1_{)2 + (}𝑑2_{)2 = (}11_{)2 + (}26_{)2 = 14.12 m}

x

x 𝑑_{1 𝑑2}

EXERCISES

1. The diagonal of a rectangle is 25 meters long and makes an angle of 36° with one side of the rectangle. Find the area and the perimeter of the parallelogram.

2. Determine the area of a rectangle whose diagonal is 24cm and the angle between the diagonals is 60°.

3. A side of a square is 16 inches. The midpoints of its sides are joined to form an inscribed square. Another square is drawn in such a way that its vertices would lie also at the midpoints of the sides of the second square. This process is

continued infinitely. Find the sum of the areas of these infinite squares.

4. A rectangle and square have the same area. If the length of the side of the square is 6 units and the longest side of the rectangle is 5 more than the measure of the shorter side, find the dimensions of the rectangle.

5. Determine the sides of the rectangle if they are in the ratio of 2 is to 5, and its area is equal to 90cm2_.

6. Find the height of a parallelogram with sides 10 and 20 inches long, and an included angle of 35°. Also, calculate the area of the figure.

7. A certain city block is in the form of a parallelogram. Two of its sides measure 32 ft. and 41ft. If the area of the land in the block is 656ft.2_{,what is the length of its} longer diagonal?

8. The area of an isosceles trapezoid is 246m2_{. If the height and the length of one of} its congruent sides measure 6m and 10m, respectively, find the lengths of the

9. An isosceles trapezoid has an area of 40m2 _{and an altitude of 2m. Its two bases} have a ratio of 2 is to 3. What are the lengths of the bases and one diagonal of the trapezoid?

10. A piece of wire of length 52m is cut into two parts. Each part is when bent to form a square. It is found that combined area of the two square is 109m2_{. Find} the measures of the sides of the two squares.

11. A rhombus has diagonals of 32 and 20 inches. Find the area and the angle opposite the longer diagonal.

12. If you double the length of the side of a square, by how much do you increase the area of that square?

13. If the diagonal length of a square is tripled, how much is the increase in the perimeter of that square?

14. If the length and width of a rectangle are doubled, by what factor is the length of its diagonal multiplied?

15. The area of the rhombus is 156m2_{. If its shorter diagonal is 13m, find the length} of the longer diagonal.

16. A garden plot is to contain 240 sq. ft. If its length is to be three time its width, what should its dimension be?

17. The altitude BE of parallelogram ABCD divides the side AD into segments in the ratio 1:3. Find the area of the parallelogram if the length of its shorter side AB is 14cm, and one of its interior angle measures 60°.

18. The official ball diamond is in the form of a square. The distance between the home base and the second base in a baseball is usually 35m. Find the area and the distances between the bases.

19. The vertical end of a trough, which is in the form of a trapezoid, has the

following dimensions: width at the top is 1.65m, width at the bottom is 1.15m, and depth is 1.35m. Find the area of this section of the trough.

20.A piece of wire is shaped to enclose an equilateral triangle in which the area is 16 3 cm2_{. It is then reshaped to enclose a rectangle whose length is 9cm. Find} the area of the rectangle.

21. A square section ABCD has one of its sides equal to x. Point E is inside the

square forming an equilateral triangle BEC with one side equal in length to the side of the square. Find angle AED.

22.A rectangle ABCD which measures 9 ft., is folded once perpendicular to diagonal AC such that the opposite vertices A and C coincide. Find the length of the fold. 23. A quadrilateral contains two sides measuring 12 cm each and an included right

angle. If the measure of the third side is 8 cm and the angle opposite the right side angle is 120°, find the measure of the fourth side and the area of the

quadrilateral.

24.The four angles of a trapezium have the same constant difference between them. If the smallest angle is 75°, find the measure of the second largest angle.

25. The distance between the center of symmetry of a parallelogram and its longer side is equal to 12 cm. The area of the parallelogram is 720 cm2_{, and its}

perimeter is 140 cm. Determine the length of the longer diagonal of the parallelogram.

26.Find the area of the rhombus in which one side measures 10 cm and a diagonal measures 12 cm.

27. The lengths of the parallel sides of an isosceles trapezoid are 8 in. and 16 in., respectively. If the diagonal bisects the base angle, what is the area of the trapezoid?

28.The perimeter of an isosceles trapezoid is 62 cm. If three sides are equal in length and the fourth side is 10 cm longer, find the area of the trapezoid.

29.The longer diagonal of a parallelogram measures 62 cm and makes an angle of 30° with the base. Find the area of the parallelogram if the diagonals intersect at an angle of 70°. Hind: use the formula 𝐴 = 1₂𝑑₁𝑑_{2𝑠𝑖𝑛𝜃}, where 𝜃 is the included angle between diagonals d₁ and d₂.

30.A diagonal of an isosceles trapezoid measures 20 in. and makes an angle of 30° with the base. If one of the congruent sides measures 15 in., find the area of the trapezoid.

Chapter Test

I. Completion of Statements

1. If three sides of one triangle are equal respectively to three sides of another, the triangles are said to be _________.

2. Corresponding parts of congruent triangles are ________.

3. If the median of a triangle is also the altitude, the triangle is _______. 4. In a right triangle, the side opposite the right angle is called _______.

5. The _______ of a triangle is the line connecting a vertex and the midpoint of the opposite side of the triangle.

6. The sum of the three angles in any triangle is _______. 7. A triangle is _______ if it has two congruent altitudes. 8. A regular polygon of three sides is called a/an _______. 9. A regular polygon of four sides is called a/an _______.

10. The sum of the measures of the angles in a quadrilateral in _______. 11. A trapezoid is said to be a/an _______. If two of its angles measure 90°. 12. The intersection of the angle bisectors of a triangle is called _______. 13. In an isosceles triangle, the _______ is located one-third of its altitude

from the base.

14. In naming of polygons, the word “kai” means _______. 15. A quadrilateral with no parallel sides is called _______.

II. True-False Statements

______ 1. A line perpendicular to another line also bisects the line. ______ 2. An equilateral triangle is also equiangular

______ 3. The altitude of a triangle always passes through the midpoint of a side. ______ 4. In an isosceles triangle, median to the base is perpendicular to the base. ______ 5. The bisector of an angle of a triangle bisects the side opposite of a side. ______ 6. The altitude of a triangle intersects the midpoint of a side.

______ 7. The bisectors of two angles of a triangle are perpendicular to each other. ______ 8. In an equilateral triangle, the altitude is a perpendicular bisector of the

base.

______ 9. In an equilateral triangle, the base angles are congruent. ______ 10. In an isosceles triangle, all three angles are acute.

______ 11. If the two diagonals of a quadrilateral are perpendicular, the quadrilateral is a parallelogram.

______ 12.A parallelogram is a rectangle. ______ 13. A square is a rectangle.

______ 14. An isosceles trapezoid has two congruent sides.

III. Place a check mark under the name of each figure that satisfies the given property.

Property Parallelogr

am

Rectangle Square Rhombus Trapezoid

All Sides are congruent Both pairs of opposite sides are parallel Both pairs of opposite sides are congruent Diagonals are congruent Diagonals bisect each other Diagonals are perpendicular