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Mastering Arcs, Chords & Tangents: Theorems & Examples

Explore the concepts of arcs, chords, and tangents in circles through theorems and examples. Learn about congruence, perpendicularity, and bisecting properties. Solve problems and master key principles in geometry.

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Mastering Arcs, Chords & Tangents: Theorems & Examples

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  1. 8-3 & 8-4 TANGENTS, ARCS & CHORDS 5 Theorems

  2. 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.

  3. THEOREM 2: • If two segments from the same exterior point are tangent to a circle, then they are congruent. tangent tangent

  4. EXAMPLES: SOLVE. (Segments that appear to be tangent are.) FIND CE and EA FIND DC

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

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