Lesson 10.1

Circles

Definition:•The set of all points in a plane that are a given distance from a given point in the plane.

•The given point is the CENTER of the circle.

•A segment that joins the center to a point on the circle is called a radius.

•Two circles are congruent if they have congruent radii.

Concentric Circles: Two or more coplanar circles with the same center.

A point is inside (in the interior of) a circle if its distance from the center is less than the radius.

interior

O

A

Point O and A are in the interior of Circle O.

A point is outside (in the exterior of) a circle if its distance from the center is greater than the radius.

AW

Point W is in the exterior of Circle A.

A point is on a circle if its distance from the center is equal to the radius.

S

Point S is on Circle A.

Chords and Diameters:

Points on a circle can be connected by segments called chords.

A chord of a circle is a segment joining any two points on the circle.

A diameter of a circle is a chord that passes through the center of the circle.

The longest chord of a circle is the diameter.

chord

diameter

Formulas to know!

Circumference:C = 2 π r

orC = π d

Area:

• A = π r2

Area:

• A = π r2

Radius-Chord Relationships

•OP is the distance from O to chord AB.

•The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord.

Theorem 74

If a radius is perpendicular to a chord, then it bisects the chord.

Theorem 75

If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.

Theorem 76

The perpendicular bisector of a chord passes through the center of the circle.

1. Circle Q, PR ST

2.PR bisects ST.

3.PR is bisector of ST.

4.PS PT

1. Given2. If a radius is to a chord, it

bisects the chord. (QR is part of a radius.)

3. Combination of steps 1 & 2.

4. If a point is on the bisector of a segment, it is equidistant from the endpoints.

The radius of Circle O is 13 mm.The length of chord PQ is 10 mm.Find the distance from chord PQ to center, O.

1. Draw OR perpendicular to PQ.2. Draw radius OP to complete a right Δ.3. Since a radius perpendicular to a

chord bisects the chord, PR = ½ PQ = ½ (10) = 5.

4. By the Pythagorean Theorem, x2 + 52 = 132

5. The distance from chord PQ to center O is 12 mm.

1. ΔABC is isosceles (AB AC)

2. Circles P & Q, BC ║ PQ3. ABC P, ACB Q4. ABC ACB5. P Q6. AP AQ7. PB CQ8. Circle P Circle Q

1. Given2. Given3. ║Lines means

corresponding s .4. .5. Transitive Property6. .7. Subtraction (1 from 6)8. Circles with radii are .