Unit 4: Arcs and Chords Keystone Geometry

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Presentation transcript:

Unit 4: Arcs and Chords Keystone Geometry

Measures of an Arc Measures of an Arc: Adjacent Arcs: List: Major Arcs 1. The measure of a minor arc is the measure of its central angle. 2. The measure of a major arc is 360 - (measure of its minor arc). 3. The measure of any semicircle is 180. Adjacent Arcs: Arcs in a circle with exactly one point in common. List: Major Arcs Minor Arcs Semicircles Adjacent Arcs

Example: Find the measure of each arc. 4x + 3x + (3x +10) + 2x + (2x-14) = 360° 14x – 4 = 360° so 14x = 364 x = 26° 4x = 4(26) = 104° 3x = 3(26) = 78° 3x +10 = 3(26) +10= 88° 2x = 2(26) = 52° 2x – 14 = 2(26) – 14 = 38°

Theorem In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

Example In circle J, find the measures of the angle or arc named with the given information: Find:

In Circle C, find the measure of each arc or angle named. Given: SP is a diameter of the circle. Arc ST = 80 and Arc QP=60. Find:

Theorem #1: In a circle, if two chords are congruent then their corresponding minor arcs are congruent. E A B C D Example:

Theorem #2: 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.

Theorem #3: 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.

Try Some: 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 bisects AB 8cm 15cm O A B D x

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 small chord to the center of Circle O. 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