University:
High School
Course:
High School Mathematics
Academic year:

2024
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88

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2
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galapi61696

10.3 Using Chords In your own words, write the meaning of each vocabulary term: chord: is the circle part created by a segment that starts/stops on the circle. diameter: is a chord that goes through the center. Theorems Theorem 10.6 Congruent Corresponding Chords Theorem In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. If: AB = CD Then: AB ≅ CD Theorem 10.7 Perpendicular Chord Bisector Theorem If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If: EG ⊥ FD Then: FH ≅ HD FG ≅ GD Theorem 10.8 Perpendicular Chord Bisector Converse If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. If: SQ ⊥ RT (SQ bisects RT) Then: SQ is a diameter Theorem 10.9 Equidistant Chords Theorem In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. If: AB ≅ CD Then: FE ≅ EG (Distances are always ⊥) 10.3 Extra Practice In Exercises 1-4, find the measure of the arc or chord in Circle O. mWX = 42° Since arcs are equal to their central angles and ∠WQX ≅ ∠ZQY (vertical angles) Then, WX = ZY YZ = 3.6 Since WX = 2(YZ) Then, WX = 2(3.6) = 7.2 WZ = 9.2 Since ∠WQZ ≅ ∠XQY (vertical angles) Then, WZ = XY = 9.2 mXY = 138° ZX is a diameter So, ∠ZQX = 180° 2x = 180° x = 90° mXY = 180° - 42° = 138° In Exercises 5 and 6, find the value of x. Since PR is bisected by diameter SQ 2x - 4.1 = x + 0.5 x = 4.6 3x - 5 = 2x + 1 x = 6 In Exercises 7 and 8, find the radius of the circle. Both chords are equidistant to the center So, the chords are congruent 5x - 2 = 3x + 2 x = 2 AC = FD 5x - 2 = 3x + 2 x = 2 a² + b² = c² 3² + 4² = r² 25 = r² r = 5 Congruent chords (20) are always equal distance to the center 4x - 5 = x + 4 3x = 9 x = 3 a² + b² = c² 7² + 10² = r² 49 + 100 = r² 149 = r² r ≈ 12.2