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### Circles

Chapter 10

10.1 Tangents to Circles

- Circle: the set of all points in a plane that are equidistant from a given point.
- Center: the given point.
- Radius: a segment whose endpoints are the center of the circle and a point on the circle.

Vocabulary

- Chord: a segment whose endpoints are points on the circle.
- Diameter: a chord that passes through the center of the circle.
- Secant: a line that intersects a circle in two points.
- Tangent: a line in the plane of a circle that intersects the circle in exactly one point.

More Vocabulary

- Congruent Circles: two circles that have the same radius.
- Concentric Circles: two circles that share the same center.
- Tangent Circles: two circles that intersect in one point.

Tangent Theorems

- If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

P

Q

Tangent Theorems

- In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

P

Q

Tangent Theorems

- If two segments from the same exterior point are tangent to a circle then they are congruent.

Q

P

Examples

- Find an example for each term:
Center

Chord

Diameter

Radius

Point of Tangency

Common external tangent

Common internal tangent

Secant

Examples

- The diameter is given. Find the radius.
- d=15cm
- d=6.7in
- d=3ft
- d=8cm

Examples

- The radius is given. Find the diameter.
- r = 26in
- r = 62ft
- r = 8.7in
- r = 4.4cm

10.2 Arcs and Chords

- An angle whose vertex is the center of a circle is a central angle.
- If the measure of a central angle is less than 180 , then A, B and the points in the interior of APB form a
minor arc.

- Likewise, if it is greater
than 180, if forms a major

arc.

Arcs and Chords

- If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.
- The measure of an arc is the same as the measure of its central angle.

Examples

- Find the measure of each arc.
- MN
- MPN
- PMN

Arc Addition Postulate

- The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
- mABC = mAB + mBC

Examples

- Find the measure of each arc.
- GE
- GEF
- GF

Theorems

- In the same circle, or in congruent circles, congruent chords have congruent arcs and congruent arcs have congruent chords.
- In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center.

Examples

- Determine whether the arc is minor, major or a semicircle.
- PQ
- SU
- QT
- TUP
- PUQ

Examples

- KN and JL are diameters. Find the indicated measures.
- mKL
- mMN
- mMKN
- mJML

Examples

- Find the value of x. Then find the measure of the red arc.

Homework

- Pg. 600 # 26-28, 37, 39, 47, 48
- Pg. 607 # 12-30 even, 32-34, 37-38

10.3 Inscribed Angles

- An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.
- The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc.

Measure of an Inscribed Angle

- If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
- mADB = ½ mAB

Find the measure of the blue arc or angle

- mQTS =
- m NMP =

Theorem

- If two inscribed angles if a circle intercept the same arc, then the angles are congruent.

Polygons and circles

- If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle, and the circle is circumscribed about the polygon.

Theorems

- If a right triangle is inscribed in a circle, then the hypontenuse is a diameter of the circle.
- B is a right angle iff AC is a diameter of the circle.

Theorems

- A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.

D, E, F, and G lie on some circle, C, if and only if

mD + mF = 180° and mE + mG = 180°

Examples

- Find the value of each variable.

More Examples

- Pg 616 # 2-8

10.4 Other Angle Relationships in Circles

- If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
- m 1 = ½ mAB
- m 2 = ½ mBCA

Examples

- Line m is tangent to the circle. Find the measure of the red angle or arc.

Examples

- BC is tangent to the circle. Find m CBD.

Theorems

- If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
- x = ½ (mPS + mRQ)
- x = ½ (106 + 174 )
- x = 140

More Examples

- Pg. 624 #2-7

10.5 Segment Lengths in Circles

- When 2 chords intersect inside of a circle, each chord is divided into 2 segments, called segments of a chord.
- When this happens, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Examples

- Find x.

Vocabulary

- PS is a tangent segment because it is tangent to the circle at an endpoint.
- PR is a secant segment and PQ is the external segment of PR.

Examples

- Find x.

Examples

- Find x.

10.6 Equations of Circles

- You can write the equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.
- Suppose the radius is r and its center is at ( h, k)
- (x – h)2 + (y – k)2 = r2
- (standard equation of a circle)

Examples

- If the circle has a radius of 7.1 and a center at ( -4, 0), write the equation of the circle.
- (x – h)2 + (y – k)2 = r2
- (x – -4)2 + (y – 0)2 = 7.12
- (x + 4)2 + y2 = 50.41

Examples

- The point (1, 2) is on a circle whose center is (5, -1). Write the standard equation of the circle.
- Find the radius. (Use the distance formula)
- .
- .
- .

- (x – 5)2 + (y – -1)2 = 52
- (x – 5)2 + (y +1)2 = 25

Graphing a Circle

- The equation of the circle is: (x + 2)2 + (y – 3)2 = 9
- Rewrite the equation to find the center and the radius.
- (x – (-2))2 + (y – 3)2 = 32
- The center is (-2, 3) and the radius is 3.

Graphing a Circle

- The center is (-2, 3) and the radius is 3.

Examples

- Do Practice 10.6C or B together.

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