90 likes | 96 Views
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,
E N D
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. tangent tangent
EXAMPLES: SOLVE. (Segments that appear to be tangent are.) FIND CE and EA FIND DC
A B E C D Theorem 3: In a circle, two chords are congruent iff their corresponding minor arcs are congruent. Example: Lesson 8-4: Arcs and Chords
D B A E C Theorem 4: In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. Example: If AB = 5 cm, find AE. Lesson 8-4: Arcs and Chords
D F C O B A E Theorem 5: In a circle, two chords are congruent if and only if they are equidistant from the center. Example: If AB = 5 cm, find CD. Since AB = CD, CD = 5 cm. Lesson 8-4: Arcs and Chords
15cm A B D 8cm O 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 x Lesson 8-4: Arcs and Chords
A B 10 cm 10 cm O C 20cm D 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. Solution: ∆EOB is a right triangle. OB (radius) = 10 cm 14 cm E x 7.1 cm Lesson 8-4: Arcs and Chords