# Definitions A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. Radius – the distance.

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Definitions A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. Radius – the distance from the center to a point on the circle Congruent circles – circles that have the same radius. Diameter – the distance across the circle through its center

Diagram of Important Terms
center

Definition Chord – a segment whose endpoints are points on the circle.

Definition Secant – a line that intersects a circle in two points.

Definition Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

Example 1 Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent diameter chord radius

Definition Tangent circles – coplanar circles that intersect in one point

Definition Concentric circles – coplanar circles that have the same center.

More definitions Interior of a circle – consists of the points that are inside the circle Exterior of a circle – consists of the points that are outside the circle

Definition Point of tangency – the point at which a tangent line intersects the circle to which it is tangent point of tangency

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

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

Definitions Minor arc – Part of a circle that measures less than 180°
Major arc – Part of a circle that measures between 180° and 360°. Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle. Note : major arcs and semicircles are named with three points and minor arcs are named with two points

Diagram of Arcs

Definitions Measure of a minor arc – the measure of its central angle
Measure of a major arc – the difference between 360° and the measure of its associated minor arc.

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

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

Right Triangles Pythagorean Theorem
Radius is perpendicular to the tangent.  < E is a right angle

Example

Example Find the measure of each arc. 70° 360° - 70° = 290° 180°

Example Find the measures of the red arcs. Are the arcs congruent?

Example 3 Find the measures of the red arcs. Are the arcs congruent?

Example 4

Definitions 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 inscribed angle intercepted arc

Measure of an Inscribed Angle Theorem
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

Example 1 Find the measure of the blue arc or angle. a. b.

Congruent Inscribed Angles Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Example 2

Definitions Inscribed polygon – a polygon whose vertices all lie on a circle. Circumscribed circle – A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

Inscribed Right Triangle Theorem
If a right triangle 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.

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

Example 3 Find the value of each variable. b. a.

Tangent-Chord Theorem
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.

Example 1

Try This!

Example 2

Interior Intersection Theorem
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.

Exterior Intersection Theorem
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 measures of the intercepted arcs.

Diagrams for Exterior Intersection Theorem

Example 3 Find the value of x.

Try This! Find the value of x.

Example 4 Find the value of x.

Example 5 Find the value of x.

Chord Product Theorem 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 product of the lengths of the segments of the other chord.

Example 1 Find the value of x.

Try This! Find the value of x.

Secant-Secant Theorem
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.

Secant-Tangent Theorem
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.

Example 2 Find the value of x.

Try This! Find the value of x.

Example 3 Find the value of x.

Try This! Find the value of x.

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