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. FIND CE and EA FIND DC (Segments that appear to be tangent are.) FIND DC FIND CE and EA

Lesson 8-4: Arcs and Chords Theorem 3: In a circle, two chords are congruent iff their corresponding minor arcs are congruent. E A B C D Example: Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Theorem 4: In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. E D A C B Example: If AB = 5 cm, find AE. Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords Theorem 5: In a circle, two chords are congruent if and only if they are equidistant from the center. O A B C D F E Example: If AB = 5 cm, find CD. Since AB = CD, CD = 5 cm. Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords 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 8cm 15cm O A B D x Lesson 8-4: Arcs and Chords

Lesson 8-4: Arcs and Chords 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: 10 cm 20cm O A B D C ∆EOB is a right triangle. OB (radius) = 10 cm 14 cm E x 7.1 cm Lesson 8-4: Arcs and Chords