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CHAPTER
6.2 Chords and Secants
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Chords
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Theorem 6.5
When two chords of circle intersect, the measure of each angle formed is one-half the sum of the measures of its intercepted arc and the arc intercepted by its vertical angle.
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4. Finding Angle Measures
a. What is the value of the variable? Solution The lines intersect inside the circle, so we find half the addition of arcs.
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More Theorems
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Bisector of an Arc
A line that divides and arc into two arcs with the same measure is called a bisector of the arc.
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7. Perpendicular Line Theorems
Thm 6.8 – A line drawn from the center of a circle perpendicular to a chord bisects the chord and the arc formed by the chord. Thm 6.9 – A line drawn from the center of a circle to the midpoint of a chord (not a diameter) or to the midpoint of the arc formed by the chord is perpendicular to the chord.
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8. Congruent Chord Theorems
Thm 6.10 – In the same circle, congruent chords are equidistant from the center of the circle. Thm In the same circle, chords equidistant from the center of the circle are congruent.
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More Theorems
Thm 6.12 – The perpendicular bisector of a chord passes through the center of the circle. Thm 6.13 – If two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other.
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Definition
A secant is a line that intersects a circle at two points.
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11. Theorem 6.14 Secants Intersecting Outside a Circle
If two secants intersect forming an angle outside the circle, then the measure of this angle is one-half the difference of the measures of the intercepted arcs.
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12. Finding Angle Measures
b. What is the value of the variable? Solution The lines intersect outside the circle, so we find half the difference of arcs.
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Finding an Arc Measure
A satellite in a geostationary orbit above Earth’s equator has a viewing angle of Earth formed by the two tangents to the equator. The viewing angle is about 17.5°. What is the measure of the arc of Earth that is viewed from the satellite? Solution The sum of the measures of the arcs of a circle is 360°. Let
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Finding an Arc Measure
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15. Theorem 6.15 Segment Products—Inside or Outside a Circle
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16. Finding Segment Lengths
Find the value of the variable in the circle. Solution 6(6 + 8) = 7(7 + y) 6(14) = 7(7 + y) 84 = y 35 = 7y 5 = y
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17. Finding Segment Lengths
Find the value of the variable in the circle. Solution 8(8 + 16) = z2 8(24) = z2 192 = z ≈ z
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