NAME _____________________________________________ DATE ____________________________ PERIOD _____________
10-5 Study Guide and Intervention
Tangents
Tangents A tangent to a circle intersects the circle in exactly one point,
called the point of tangency. There are important relationships involving
tangents. A common tangent is a line, ray, or segment that is tangent to two
circles in the same plane.
• A line is tangent to a circle if and only if it is perpendicular to a radius at a
point of tangency.
• If two segments from the same exterior point are tangent to a circle, then
they are congruent.
̅̅̅̅ ⊥ ̅̅̅̅
̅̅̅̅ is
If 𝑅𝑆
𝑅𝑃, then 𝑆𝑅
̅̅̅̅ is tangent
tangent to ⨀P. If 𝑆𝑅
̅̅̅̅ ⊥ ̅̅̅̅
to ⨀P, then 𝑅𝑆
𝑅𝑃.
̅̅̅̅ and 𝑆𝑇
̅̅̅̅ are tangent to ⨀P,
If 𝑆𝑅
̅̅̅̅ ≅ 𝑆𝑇
̅̅̅̅.
then 𝑆𝑅
̅̅̅̅ is tangent to ⨀C. Find x.
Example: 𝑨𝑩
̅̅̅̅ is perpendicular to radius 𝐵𝐶
̅̅̅̅ . 𝐶𝐷
̅̅̅̅ is a radius, so CD = 8
AB is tangent to ⨀C, so 𝐴𝐵
and AC = 9 + 8 or 17. Use the Pythagorean Theorem with right △ABC.
(𝐴𝐵)2 + (𝐵𝐶)2 = (𝐴𝐶)2
2
2
𝑥 + 8 = 17
2
2
𝑥 + 64 = 289
𝑥 2 = 225
x = 15
Pythagorean Theorem
Substitution
Simplify.
Subtract 64 from each side.
Take the positive square root of each side.
Exercises
Find x. Assume that segments that appear to be tangent are tangent.
1.
2.
3.
4.
5.
6.
Chapter 10
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Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
10-5 Study Guide and Intervention (continued)
Tangents
Circumscribed Polygons When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to
the circle.
Hexagon ABCDEF is circumscribed about ⨀P.
̅̅̅̅
𝐴𝐵, ̅̅̅̅
𝐵𝐶 , ̅̅̅̅
𝐶𝐷, ̅̅̅̅
𝐷𝐸, ̅̅̅̅
𝐸𝐹 , and ̅̅̅̅
𝐹𝐴 are tangent to ⨀P.
Square GHJK is circumscribed about ⨀Q.
̅̅̅̅
̅̅̅, ̅̅̅
𝐺𝐻, ̅𝐽𝐻
𝐽𝐾 , and ̅̅̅̅
𝐾𝐺 are tangent to ⨀Q.
Example: △ABC is circumscribed about ⨀O. Find the perimeter of △ABC.
△ABC is circumscribed about ⨀O, so points D, E, and F are points of tangency.
Therefore AD = AF, BE = BD, and CF = CE.
P = AD + AF + BE + BD + CF + CE
= 12 + 12 + 6 + 6 + 8 + 8
= 52
The perimeter is 52 units.
Exercises
For each figure, find x. Then find the perimeter.
1.
2.
3.
4.
5.
6.
Chapter 10
30
Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
10-5 Skills Practice
Tangents
Determine whether each segment is tangent to the given circle. Justify your answer.
̅̅̅̅
1. 𝐻𝐼
̅̅̅̅
2. 𝐴𝐵
Find x. Assume that segments that appear to be tangent are tangent. Round to the nearest tenth if necessary.
3.
4.
5.
6.
For each figure, find x. Then find the perimeter.
7.
Chapter 10
8.
30
Glencoe Geometry