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8-3 & 8-4 TANGENTS, ARCS & CHORDS 5 Theorems THEOREM 1: a line that intersects a circle is tangent to a circle IFF it is perpendicular to the radius drawn to the point of tangency. THEOREM 2: If two segments from the same exterior point are tangent to a circle, then they are congruent. EXAMPLES: SOLVE. (Segments that appear to be tangent are.) A 1.8 cm B C FIND DC F 7.0 cm 2.4 cm D FIND CE and EA E Theorem 3: In a circle, two chords are congruent iff their corresponding minor arcs are congruent. A If AB CD then AB CD E C Example: B D Given mAB 127 , find the mCD. Since m AB mCD mCD 127 Lesson 8-4: Arcs and Chords 5 Theorem 4: In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. If DC AB then DC bi sec ts AB and AB. AE BE and AC BC D Example: If AB = 5 cm, find AE. If mAB 120 , find mAC. AE A E B C AB 5 AE 2.5 cm 2 2 m AB 120 m AC , m AC 60 2 2 Lesson 8-4: Arcs and Chords 6 Theorem 5: In a circle, two chords are congruent if and only if they are equidistant from the center. D F CD AB iff OF OE Example: If AB = 5 cm, find CD. C O A E B Since AB = CD, CD = 5 cm. Lesson 8-4: Arcs and Chords 7 Try Some Sketches: Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle. Draw a radius so that it forms a right triangle. How could you find the length of the radius? Solution: ∆ODB is a right triangle and OD bi sec ts AB AB 15 DB= = =7.5 cm 2 2 OD=8 cm OB2 =OD 2 +DB2 OB2 =82 +(7.5) 2 =64+56.25=120.25 OB= 120.25 11cm Lesson 8-4: Arcs and Chords A 15cm D 8cm B x O 8 Try Some Sketches: Draw a circle with a diameter that is 20 cm long. Draw another chord (parallel to the diameter) that is 14cm long. Find the distance from the smaller chord to the center of the circle. AB 14 7cm Solution: OE bi sec ts AB. EB 2 2 14 cm ∆EOB is a right triangle. OB (radius) = 10 cm A B E OB 2 OE 2 EB 2 x 10 X 7 2 2 10 cm 2 C 20cm 10 cm O D X 2 100 49 51 X 51 7.1 cm Lesson 8-4: Arcs and Chords 9