1 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 2 Diagram of Important Terms 3 Definition Chord – a segment whose endpoints are points on the circle. 4 Definition Secant – a line that intersects a circle in two points. 5 Definition Tangent – a line in the plane of a circle that intersects the circle in exactly one point. 6 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 7 Definition Tangent circles – coplanar circles that intersect in one point 8 Definition Concentric circles – coplanar circles that have the same center. 9 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 10 Example 2 Tell whether the common tangents are internal or external. 11 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 12 Definition Point of tangency – the point at which a tangent line intersects the circle to which it is tangent point of tangency 13 Perpendicular Tangent Theorem 14 Perpendicular Tangent Converse 15 Definition central angle 16 Definitions Minor arc – Part of a circle that measures less than 180° 17 Diagram of Arcs 18 Definitions Measure of a minor arc – the measure of its central angle 19 Arc Addition Postulate 20 Definition Congruent arcs – two arcs of the same circle or of congruent circles that have the same measure 21 Arcs and Chords Theorem 22 Perpendicular Diameter Theorem 23 Perpendicular Diameter Converse 24 Congruent Chords Theorem 25 Example 3 Use the converse of the Pythagorean Theorem to see if the triangle is right. ? 452 ? 2025 1970 2025 26 Congruent Tangent Segments Theorem 27 Example 4 28 Example 1 Find the measure of each arc. 70° 360° - 70° = 290° 180° 29 Example 2 Find the measures of the red arcs. Are the arcs congruent? 30 Example 3 Find the measures of the red arcs. Are the arcs congruent? 31 Example 4 32 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 33 Measure of an Inscribed Angle Theorem 34 Example 1 Find the measure of the blue arc or angle. a. b. 35 Congruent Inscribed Angles Theorem 36 Example 2 37 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. 38 Inscribed Right Triangle Theorem 39 Inscribed Quadrilateral Theorem 40 Example 3 Find the value of each variable. b. a. 41 Tangent-Chord Theorem 42 Example 1 43 Try This! 44 Example 2 45 Interior Intersection Theorem 46 Exterior Intersection Theorem 47 Diagrams for Exterior Intersection Theorem 48 Example 3 Find the value of x. 49 Try This! Find the value of x. 50 Example 4 Find the value of x. 51 Example 5 Find the value of x. 52 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. 53 Example 1 Find the value of x. 54 Try This! Find the value of x. 55 Secant-Secant Theorem 56 Secant-Tangent Theorem 57 Example 2 Find the value of x. 58 Try This! Find the value of x. 59 Example 3 Find the value of x. 60 Try This! Find the value of x. |