Unit 8: Circle Geometry
Necessary Prior Knowledge
A circle is a set of points that are a fixed distance from a fixed point. The fixed point is the center of the circle, and is often labeled O. The fixed distance is the radius of the circle.
An angle is formed when the endpoints of two line segments meet. The point where they meet is called the vertex of the angle. Angles are named with 3 uppercase letters of the alphabet, beginning with one endpoint and ending with the other endpoint. The vertex of the angle should always be in the center. The angle below can be named or .
A line or line segment is named with two uppercase letter of the alphabet. A point is named with only one uppercase letter of the alphabet.
An Isosceles triangle has two equal sides and two equal angles.
The sum of the angles of a triangle is 1800.
The Pythagorean Theorem can be used to find the length of a missing side of a right triangle.
Unit 8: Circle Geometry A
O B
O
.
8.1. Properties of Tangents to a Circle
Recall that perpendicular lines meet at a right angle or 900.
A tangent is a line or segment that touches a circle at a single point. That point is referred to as the point of tangency. Note that there is only one tangent that can be drawn at any point.
Investigate:
If we were to take a large circle, draw a radius and then a tangent to the circle where the radius touches the circle, what would be the angle between the radius and the
tangent line?
Tangent-Radius Property:
A tangent to a circle is perpendicular to the radius at the point of tangency.
O
.
•
A•
P•
BTangent
Example 1 p. 386
Recall that the Pythagorean Theorem is used to find a missing side in a right triangle.
If we are looking for the hypotenuse we use the addition form: (Hypotenuse)2 = (leg 1)2 + (leg 2)2
If we are looking for a leg we can use the subtraction form: (leg 1)2 = (Hypotenuse)2 - (leg 2)2
Problem solving
Example 3 p. 387
Additional Example: Mike was flying a remote-control airplane in a circle with a radius of 50 m. The signal was lost by the airplane which then flew along a tangent from the circle until it crashed 140 m from Mike’s location at the centre of the circle. How far did the airplane travel horizontally along the tangent to the nearest meter?
Solution:
The airplane travelled about 131 m along the tangent.
Recall that the radii of a circle are equal in length. If we were to draw tangents from the endpoints of two radii so that they meet, their lengths would be the same. This can help us find missing lengths as well.
See the diagram in #16 p. 390.
Note that x and y are the same length and we can find either one using the Pythagorean Theorem.
O 50 m
Complete pp. 388-389 #3-6 orally together
8.2. Properties of Chords in a Circle
A chord is a line segment with endpoints on the circle.
The diameter is a special chord that passes through the center of the circle.
Recall that to bisect means to cut something into two equal halves.
A perpendicular bisector cuts a line segment into two equal halves at 900.
Unit 8: Circle Geometry
Investigate:
If we were to take a large circle and draw two different chords, we could fold the paper to construct the perpendicular bisector of each chord. If we then labeled the point inside the circle where the two perpendicular bisectors intersect, what would you notice?
Perpendicular to Chord Property:
If any two of the following conditions are in place, the third condition will also be true:
The line bisects a chord
The line goes through the centre of the circle
The line is perpendicular to the chord.
Note we can use this property to find the center of a circle. See p. 399 #16 below:
Recall again, that the radii of a circle are equal in length. If we create a triangle with two radii and a chord the triangle will be isosceles and therefore have two equal angles.
Example 2 p. 395
Example 3 p. 396
See p. 397 #3-5 & p. 398 #11 orally together:
Set pp. 397-399 #6, 7, 9, 10*, 12, 14, 15a, 17, 18
*Might be tough for some students
8.3. Properties of Angles in a Circle
A central angle is an angle with its endpoints on the circle and its vertex at the center of the circle. The line segments that form the angle are radii of the circle. is a central angle in the diagram below left.
An inscribed angle is an angle with its endpoints and vertex on the circle. The line segments that form the angle are chords of the circle. is an inscribed angle in the diagram above right.
Also see diagram bottom p. 405
An inscribed polygon is a polygon whose vertices lie on a circle.
Unit 8: Circle Geometry
A
B
O C
An arc is part of the circumference of the circle. It is named with two
uppercase letters, the endpoints of the arc. Note that there are always two arcs, a major arc (the longer one) and a minor arc (the shorter one).
An angle that subtends an arc is an angle that is formed by the endpoints of the arc. Note: In the diagram at the bottom of p. 405 is a central angle subtended on the arc AB and is an inscribed angle subtended on the arc AB.
Since the inscribed angle is half the central angle subtended on the same arc, and the diameter is a straight angle (1800), the inscribed angle on the
same arc must be half of 1800, or 900.
Examples: Find the indicated measures in the diagrams below.
a)
b) x and y
Unit 8: Circle Geometry
Solution:
is 220 since
inscribed angles subtended by the same arc are equal.
Solution:
y is 560 since inscribed
angles subtended by the same arc are equal.
x is 1120 since a
c) x, and
Example 1 p. 407:
Solution:
Since inscribed angles subtended by the same arc are equal
.
Solving this equation gives .
Since and
are the same we can evaluate either for
.
Example 2 pp. 407-408:
See p. 410 #3 & 4:
Discuss #10 p. 411:
Set pp. 410-412 #5, 6, 9, 11
