b d b d Cyclic
Expression
(a + b) 2 = a2 + b2 + 2 a ; (a - b) 2 = a2 + b2 - 2 a b;
(a - b) 2 = a2 + b2 - 2 a b; a2 - b2 = (a + b) x (a - b).
a3 – b3 = (a - b) (a2 + b2 + a b) ; a3 + b3 = (a + b) (a2 + b2 - a b);
(a + b) 3 = a3 + b3 + 3 a b (a + b);
(a - b) 3 = a3 - b3 - 3 a b (a - b).
G EOM ETRY
Geometry: Geometry is all about shapes and their properties. Geometry can be divided into two parts.
Plane Geometry: Plane Geometry is about flat shapes like line, plane, triangle, Quadrilateral and circles that can be drawn on a piece of paper
Triangle: Triangles are assumed to be two-dimensional plane figures. A triangle is one of the basic shape of Geometry or a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC. The three angles always add to 180°. A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. A "triangle" with an interior angle of 180° and collinear vertices is degenerate.
Triangle Shapes
Right Angle Triangle:
A right triangle has one of its interior angles measuring 90°. The side opposite to the right angle is the hypotenuse;
it is the longest side of the right triangle. The other two sides are called the legs of the triangle.
Scalene Triangle:
Scalene Triangle has no equal sides and no equal angles. Obtuse Triangle has all three angles less than 90°.
Equilateral Triangle: In
an equilateral triangle, all sides have the same length. In equilateral triangle is also a regular polygon with all angles measuring 60°.
Isosceles Triangle:
Isosceles triangle has two sides equal in length and two angles opposite to the two sides of the same length have same measure.
Obtuse Angle
Triangle: Obtuse Angle Triangle has an angle more than 90°
Oblique Triangles:
Triangles that has all sides different and do not have an angle that measures 90° are called oblique triangles.
In diagrams representing triangles above, "tick" marks are used to denote sides of equal lengths, such as, the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.
Area of Triangles: The area of a triangle can be demonstrated as half of the area of a parallelogram which has the same base length and height. Simplest formula is:
Where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term 'base' denotes any side and 'height' denotes the length of a perpendicular from the vertex opposite the side onto the line containing the side itself. The sides of the triangle are known as follows: The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h. The opposite side is the side opposite to the angle we are interested in, in this case a. The adjacent side is the side that is in contact with the angle we are interested in.
Heron's formula: The shape of the triangle is determined by the lengths of the sides alone. Therefore the area can also be derived from the lengths of the sides. By Heron's formula:
Where “s” = half of the triangle's perimeter. Three other way of finding the Triangle area by Heron's formula is:
Quadrilaterals: Quadrilateral has a four-sided two-dimensional shape. The sides are straight and the interior angles add up is equal to 360 degrees. There are many types of quadrilateral:
Rectangle: A rectangle is a four-sided shape where every angle is a right angle (90°) and opposite sides are parallel and of equal length.
Rhombus: A rhombus is a four-sided shape where all sides have equal length and opposite sides are parallel and opposite angles are equal. The diagonals of a rhombus bisect each other at right angles.
The Square: A square has equal sides and every angle is a right angle (90°) and opposite sides is parallel.
The Parallelogram: Opposite sides are parallel and equal in length, and opposite angles are equal.
Note: Squares, Rectangles and Rhombuses are all Parallelograms.
Trapezoid: A trapezoid (a trapezium) has one pair of opposite sides parallel. It is called an isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal.
Circle: Circle is the set of all points on a plane that are at a fixed distance from a centre. It is a round figure.
Radius and Diameter of a Circle: The Radius is the distance from the centre to the edge of the circle. The Diameter is the linear distance passing through the centre between two points on the circle, which are opposite to each other. So the Diameter is twice the Radius: Diameter = 2 × Radius.
Menstruation: Menstruation is the branch of geometry dealing with measurement of geometric magnitudes such as length, area and volume.
Area of a Triangle: The area of a Triangle is half of the base times height. Area = ½ b × h Where, b = base; & h = vertical height of a, b, c sides of triangle. Or,
Area = √s(s-a)(s-b)(s-c); where 2s = a + b + c
Area of Squire: Square Area = a x a = a2. Where, a = length of side of the Squire. Chord. If that line passes through the centre it is called a Diameter. If a line "just touches" the circle as it passes it is called a Tangent. And a part of the circumference is called an Arc. The slice made by a chord is called a Segment. Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle.