Name:______Date:______Geometry Essentials
Chapter 13: Circle Geometry BASIC TERMS
Circle – the set of all points in a plane that are a given distance from a fixed point called the center of the circle.
Radius – a line segment from the center to any point on the circle.
Theorem – All radii of the circle are congruent.
Chord – a line segment that joins any two points on the circle.
Diameter – a chord that passes through the center of the circle. (D = 2r)
Tangent – a line in the plane of the circle that intersects the circle and one and only one point (the point of tangency).
Secant – a line in the plane of the circle that intersects the circle at two points.
ARCS
Arc of a Circle – any curved section of the circle. The sum of all the non-overlapping arcs around the circle must total 360°
Semicircle – an arc that represents half the circle and measures 180°
Minor Arc – an arc whose measure is less than 180°
Major Arc – an arc whose measure is greater than 180°
ANGLES
Central Angle – of circle is an angle whose vertex is the center of the circle and whose sides are radii
*The degree measure of an arc is equal to the measure of the central angle that intercepts the arc.
Congruent Circles – are circles with congruent radii.
Congruent Arcs – are arcs in the same circle or in congruent circles that have the same degree measure.
Theorem – In a circle or congruent circles, congruent central angles intercept congruent arcs, and conversely, congruent arcs are intercepted by congruent central angles.
Example: In circle O, AC and BE are diameters. m B (a) m (b) m (c) m (d) m (e) m AB = (f) m BAD = D (g) m DBE = E Tangents and Chords THEOREMS ABOUT TANGENT LINES TO CIRCLES Theorem 1: The radius of a circle is ______to the tangent line at the point of tangency Theorem 2: Two tangents to a circle from the same external point are ______ COMMON TANGENTS Use the diagrams below to state the number of Common Tangents between two circles: *______* *______* *______* *______* *_____* CHORDS *If a diameter is perpendicular to a chord, then the diameter bisects the chord *If a diameter bisects the chord, then it is perpendicular to the chord. Theorem: If two chords intersect within a circle, the product of the measures of the segments of one chord equals the product of the measures of the segments of the other. ______ Example 1: In circle O, chords AB and CD intersect at point E. If AE = 6, EB = 8, and CE = 4, find the length of ED. Example 2: In circle O, chords RN and AB intersect at E. If AB = 16, RE = 6, and EN = 8, find the length of AE given that AE < EB. Example 3: Find the length of a chord 4 cm from the center of a circle whose radius measures 5cm. Example 4: A chord 4√15 long is 2cm from the center of the circle. Find the length of the radius of the circle. Tangents and Secants Example 1: Circle O is inscribed in triangle MNP with tangent point A, B and C. If MN = 12, MP = 14 and NB = 7, find the perimeter of the triangle. M C A P N B Theorem(Tangents and Secants): If a tangent and a secant are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the whole secant and its external segment. (PB)2 = (PC) (PA) B P C A Example 2: Secant ABC and tangent AD are drawn to circle O from external point A. If AD = 8 and BC = 12, find the length of AC. C B A D Theorem (Two Secants): If two secants intersect outside a circle, then the product of the measures of the whole secant and external segment is equal to the product of the measure of the other whole secant and its external segment (PA) (PB) = (PD) (PC) A B P C D Example 3: In circle O, secants PBA and PCD intersect at external point P. If PA = 12, PB = 5 and PD = 15, find the length of PC. P B C A D Arcs and Angles of Circles Theorem – In a circle, congruent arcs have congruent corresponding chords. Conversely, congruent chords have congruent corresponding arcs. B C If AB ≅ CD, then AB͡ ≅ CD͡ Theorem – Parallel lines that intersect a circle intercept equal arcs. B A D If AB ∥ CD, then AD͡ ≅ B͡C A C D Central Angles – of a circle is an angle whose vertex is the center of the circle and whose sides are radii. **The measure of a central angle is equal to the measure of its intercepted arc A O m B Inscribed Angle – of a circle is an angle whose vertex is on the circle and whose sides contain chords. **The measure of an inscribed angle is equal to one-half of the measure of its intercepted arc. A B 1 m C Theorem: Inscribed angles that intercept the same arc are congruent. B D C m AR͡ = 80° m< ADR = _____ m m< ACR = _____ R A Example 1: Triangle ART is inscribed in circle O with m RT = 86° R