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CHAPTER
6.3 Tangents
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Congruent Arcs
Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.
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Congruent Arcs
It is possible for two arcs of different circles to have the same measure but different lengths.
It is also possible for two arcs of different circles to have the same length but different measures.
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Congruent Arcs
Arcs not congruent: • same measure • not same or congruent circles, so different lengths • same lengths • not same or congruent circles, so different measures
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Tangents
A tangent to a circle is a line in the
plane of the circle that intersects the
circle in exactly one point.
The point where a circle and a tangent
intersect is the point of tangency.
is a tangent line.
is a tangent ray.
is a tangent segment.
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Postulate 6.3
If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.
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Postulate 6.4
If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.
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8. Finding Angle Measures
Multiple Choice and are tangent to circle O. What is the value of x? a. 58 b. 63 c. 90 d. 117 Solution A tangent line is perpendicular to the radius at the point of tangency. The two given segments are tangent to the circle. Angles L and N are right angles. LMNO is a quadrilateral. The sum of the angles is 360 degrees.
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9. Finding Angle Measures
m∠L + m∠M + m∠N + m∠O = 360° 90° + m∠M + 90° + 117° = 360° 297° + m∠M = 360° m∠M = 63° Since m∠M = 63°, or x°, then x = 63. The correct answer is b.
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Finding a Radius
What is the radius of circle C? Solution Use the Pythagorean Theorem to find x.
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Theorem 6.19
If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.
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Construction
Construction 6.1 Construction 6.2 Page 299
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Theorem 6.16
The angle formed by a tangent and a chord has a measure one-half the measure of it’s intercepted arc.
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Theorem 6.17
The angle formed by the intersection of a tangent and a secant has measure one-half the difference of the measures of the intercepted arcs.
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Theorem 6.18
The angle formed by the intersection of two tangents has a measure one half the distance of the intercepted arcs.
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Theorem 6.20
If a secant and a tangent are drawn to a circle from an external point, the length of the tangent segment is the geometric mean between the length of the secant and its external segment.
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17. Finding Tangent Segment Lengths
If are tangents to circle D, find the value of x. Solution From Theorem , we know that AC ≅ AB.
The value of x is 9 or 1.
The check is left to you.
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Tangents
Two circles in a plane can share a tangent line. Common tangents are lines or segments or rays that are tangent to more than one circle.
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19. Possible Intersections of Two Circles Two Points of Intersection
2 Common Tangents 2 external tangents (green) 0 internal tangents
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20. Possible Intersections of Two Circles One Point of Intersection
3 Common Tangents 2 externally tangent circles 2 external tangents (green) 1 internal tangent (black)
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21. Possible Intersections of Two Circles One Point of Intersection
1 Common Tangent 2 internally tangent circles 1 external tangent (green) 0 internal tangent
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22. Possible Intersections of Two Circles No Points of Intersection
0 Common Tangents 2 concentric circles one circle floating inside the other, without touching 0 external tangents 0 external tangents 0 internal tangents 0 internal tangents
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23. Possible Intersections of Two Circles No Points of Intersection
4 Common Tangents 2 external tangents (green) 2 internal tangents (black)
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Tangents
The line that passes through the centers of two circles is called their line of centers. If two circles are tangent, internally or externally (see middle examples above), then their common point of tangency is on their line of centers, and these circles are called tangent circles.
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Constructions
Page 307 Construction 6.3 Construction 6.4
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