Chapter 10 Properties of Circles

10.1 Using Properties of Tangents Circle- a set of all points in a plane that are equidistant from a given point called the center

Radius- a segment whose endpoints are the center and any point on the circle Chord- a segment whose endpoints are on a circle Diameter- a chord that contains 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

Can you name it? Chord Radius Diameter Secant Tangent Point of Tangency

Coplanar circles Concentric circles Internally tangent circles Externally tangent circles

Common tangents Internal common tangent External common tangent

Theorems In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

Tangent segments from a common external point are congruent.

Examples Is segment BC tangent to circle A if segment AB is a radius?

Example S is a point of tangency. Find r.

Example Point R and T are tangent to circle P. Find x.

Example How many common tangents?

10.2 Finding Arc Measures Central angle- an angle whose vertex is the center of the circle Major arc Minor arc Semicircle

Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Congruent Circles and Arcs Two circles are congruent if they have the same radius. Two arcs are congruent if they have the same measure and they are arcs of the same circle or of congruent circles. The radii of a circle, or of congruent circles, are congruent.

Examples

Example Are arcs AB and DE congruent?

Example Ages of people in a town (in years)

10.3 Applying Properties of Chords In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Example: Find the measure of arc SR.

If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Example

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

Example BC= 2x +6 ED = 3x – 1 Find BC

Example Three props are placed on a stage (P,Q,R). Where do you put the table so that it is the same distance from each prop?