Chapter 10 – Circles 10.1 Circles and Circumference

Circle – the locus or set of all points in a plane that are

equidistant from a given point, called the center

When naming a circle you always name it by its center.

Ex. Circle A

Radius – the distance from the center to a point on the circle

The radius is half of the diameter.

Diameter – the distance across a circle, through the center

The diameter is twice the radius.

Chord – a segment whose endpoints are points on the circle

Concentric circles – coplanar circles that have a common center

Example: Circle all of the segments that are chords.

𝑀𝑁̅̅ ̅̅ ̅ 𝑀𝑂̅̅ ̅̅ ̅ 𝐿𝑀̅̅ ̅̅ ̅ 𝐿𝑁̅̅ ̅̅

Name the circle: ______________

Name a radius: ________

Name a diameter: __________

Finding Measures in Intersecting Circles

Example: The diameter of Circle S is 26 units, the radius of Circle R

is 14 units, and DS = 5 units. Find CD.

Circumference: the distance around the circle Example: Find the circumference of circle A and circle B.

O

M

N

L

A

C=2πr

C=πd

A B

a. b.

Example: If the circumference of a circle is 65.4 feet find the diameter and radius. Round to the nearest hundredth.

Area: the amount of space inside the circle Example: Find the area of circle A and circle B. Example: If the area of a circle is 256 feet, find the diameter. Round to the nearest hundredth.

Inscribed Circle – the largest possible circle that can be drawn inside a polygon Each side of the polygon is tangent.

Circumscribed Circle – the circle that can be drawn outside a polygon passing through all the vertices of the polygon

A=πr2

A B

a. b.

10.2 Measuring Arcs and Angles

Central angle – an angle whose vertex is the center of a circle

Arc measure – the same as its corresponding central angle

Sum of Central Angles – the sum of the central angles in a

circle is 360°

Minor arc – part of a circle that measures less than 180°

Smaller than semicircle.

Major arc – part of a circle that measures between 180° and 360°

Smaller than semicircle.

Semicircle – an arc whose endpoints are the endpoints of a

diameter of the circle

A semicircle measures exactly 180°.

Example: In circle E, find the measure of the angle or the arc named.

1

2

BC

AC

ADB

Theorem 10.1: In the same circle, or in congruent

circles, two minor arcs are congruent if and only if their

corresponding chords are congruent.

Arc Addition Postulate: The measure of an arc

formed by two adjacent arcs is the sum of the measures

of the two arcs.

AD BC

Ex.

Ex.

AD DB ADB

< 180

> 180

= 180

Central Angle = Measure of Arc

Example: In circle C with diameter SP, find the measure of each angle and classify each arc as a major arc, minor arc, or semicircle then find its measure.

Arc Length: the distance between the endpoints of an arc

Example: Find the length of the PQ . Round to the nearest hundredth.

Example: Find the length of the BDC . Round to the nearest hundredth.

ST

SQP

SQ

SPQ

SPT

TSQ

PT

PCQ

SCQ

SCP

TCP

2360

xl r

4

10.3 Arcs and Chords

Example: Find the value of x in each circle.

Example: Find the value of x in each circle.

Example: Find the value of x in each circle.

Theorem 10.2: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Theorem 10.3: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Theorem 10.4: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

Theorem 10.5: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

TS RS

Ex. AB CD

Ex. TU UR

AB CD

Ex. QS is the

diameter

Ex.

a. b.

a.

a.

b.

b.

10.4 Inscribed Angles

Inscribed angle – an angle whose vertex is on a circle and

whose sides contain chords of the circle

Intercepted arc – the arc that lies in the interior of an inscribed

angle and has endpoints on the angle.

Measure of an Inscribed Angle:

]

Example: Find x.

Example: Find DC and ADB Example: Find the values of x, y, and z.

Example: Find x.

Theorem 10.6: If an angle is inscribed in a

circle, then its measure is half the measure of its intercepted arc.

Theorem 10.7 : If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

A

B

C

D

A B

Ex.

Inscribed polygon – drawn inside. A polygon is inscribed in a circle if all its vertices lie on the circle. Example: Find the value of x. Example: Find the value of x. Example: Find the value of x and y.

Theorem 10.8 : If a right triangle is inscribed is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

Theorem 10.9 : A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Ex.

10.5 Tangents

Tangent – a line in the plane of a circle that intersects

the circle in exactly point.

Point of Tangency – the point at which the tangent

line intersects the circle.

2 TYPES:

Common External Tangents Common Internal Tangents

Example: Find the value of x.

Example: Find the value of x.

Theorem 10.10 : If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Theorem 10.11 : If two segments from the same exterior point are tangent to a circle, then they are congruent.

Ex.

a. b.

Ex.

a. b.

Example: Find the x then find the perimeter of the polygon.

10.6 Secants, Tangents, & Angle Measures

Secant – a line that intersects a circle in points.

Example: Find the value of x.

Example: Find the value of x.

Ex.

Theorem 10.13: 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.

Theorem 10.12: 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.

b. a.

a. b.

x

Example: Find the value of x.

Example: Find the value of x.

Theorem 10.14: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measure of the intercepted arcs.

b. a.

a.

b.

10.7 Special Segments in a Circle

Example: Find the value of x.

Example: Find the value of x.

Example: Find the value of x. Example: Find the value of y.

Theorem 10.15: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the products of the lengths of the segments of the other chord.

a. b.

Theorem 10.16: If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

Theorem 10.17: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

a. b.

10.8 Equations of Circle

Standard Form

Example: Find the equation of the circle with a center at (1, -8) and a radius of 7 and graph.

Example: Find the equation of the circle with a center at (-3, 6) that passes through the point (0, 6) and graph.

2 2 2( ) ( )x h y k r Center = (h, k) Radius = r