Circles Chapter 10

Essential Questions How do I identify segments and lines related to circles? How do I use properties of a tangent to a circle?

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.

Definitions Common tangent – a line or segment that is tangent to two coplanar circles Common internal tangent – intersects the segment that joins the centers of the two circles Common external tangent – does not intersect the segment that joins the centers of the two circles

Example 2 Tell whether the common tangents are internal or external. b. common internal tangents common external tangents

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.

Perpendicular Tangent Converse 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.

Congruent Tangent Segments Theorem If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example

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.

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

Definition Congruent arcs – two arcs of the same circle or of congruent circles that have the same measure

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.

Perpendicular Diameter Converse If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

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

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

Example 3 Use the converse of the Pythagorean Theorem to see if the triangle is right. 112 + 432 ? 452 121 + 1849 ? 2025 1970  2025

Congruent Tangent Segments Theorem If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example 4

Example 1 Find the measure of each arc. 70° 360° - 70° = 290° 180°

Example 2 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.

Inscribed Quadrilateral Theorem 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.