2 Essential QuestionsHow do I identify segments and lines related to circles?How do I use properties of a tangent to a circle?
3 DefinitionsA 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 circleCongruent circles – circles that have the same radius.Diameter – the distance across the circle through its center
5 DefinitionChord – a segment whose endpoints are points on the circle.
6 DefinitionSecant – a line that intersects a circle in two points.
7 DefinitionTangent – a line in the plane of a circle that intersects the circle in exactly one point.
8 Example 1Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.tangentdiameterchordradius
9 DefinitionTangent circles – coplanar circles that intersect in one point
10 DefinitionConcentric circles – coplanar circles that have the same center.
11 DefinitionsCommon tangent – a line or segment that is tangent to two coplanar circlesCommon internal tangent – intersects the segment that joins the centers of the two circlesCommon external tangent – does not intersect the segment that joins the centers of the two circles
12 Example 2 Tell whether the common tangents are internal or external. b.common internal tangentscommon external tangents
13 More definitionsInterior of a circle – consists of the points that are inside the circleExterior of a circle – consists of the points that are outside the circle
14 DefinitionPoint of tangency – the point at which a tangent line intersects the circle to which it is tangentpoint of tangency
15 Perpendicular Tangent Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
16 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.
17 DefinitionCentral angle – an angle whose vertex is the center of a circle.central angle
18 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
34 DefinitionsInscribed angle – an angle whose vertex is on a circle and whose sides contain chords of the circleIntercepted arc – the arc that lies in the interior of an inscribed angle and has endpoints on the angleinscribed angleintercepted arc
35 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.
36 Example 1Find the measure of the blue arc or angle.a.b.
37 Congruent Inscribed Angles Theorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
39 DefinitionsInscribed 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.
40 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.
41 Inscribed Quadrilateral Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
47 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.
48 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.
54 Chord Product TheoremIf 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.
57 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.
58 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.