A polygon is circumscribedabout a circle if each side of the polygon is tangent to the circle. When a polygon is circumscribed about a circle, we also say that the cir-cle is inscribedin the polygon. For example, in the dia-gram, is tangent to circle O at E, is tangent to circle O at F, is tangent to circle O at G, is tangent to circle O at H. Therefore, ABCD is circum-scribed about circle O and circle O is incircum-scribed in quadrilateral ABCD.
If ABC is circumscribed about circle O, then we know that , , and are the bisectors of the angles of ABC, and O is the point at which the angle bisectors of the angles of a triangle intersect.
EXAMPLE 3
, , and are tangent to circle O at D, E, and F, respectively. If AF 6, BE 7, and CE 5, find the perimeter of ABC.
Solution Tangent segments drawn to a circle from an exter-nal point are congruent.
AD AF 6 BD BE 7 CF CE 5
Tangents and Secants 563
O
EXAMPLE 4
Point P is a point on a line that is tangent to circle O at R, P is 12.0 centimeters from the center of the circle, and the length of the tangent segment from P is 8.0 centimeters.
a. Find the exact length of the radius of the circle.
b. Find the length of the radius to the nearest tenth.
Solution a. P is 12 centimeters from the center of circle O; OP 12.
The length of the tangent segment is 8 centimeters; RP 8.
A line tangent to a circle is perpendic-ular to the radius drawn to the point of tangency;OPR is a right triangle.
RP2 OR2 OP2 82 OP2 122 64 OP2 144
OP2 80 OP
b. Use a calculator to evaluate .
ENTER
:
4 5DISPLAY
:
8.94427191To the nearest tenth, OP 8.9.
Answers a. 4 cm b. 8.9 cm
Writing About Mathematics
1. Line l is tangent to circle O at A and line m is tangent to circle O at B. If is a diame-ter, does l intersect m? Justify your answer.
2. Explain the difference between a polygon inscribed in a circle and a circle inscribed in a polygon.
AOB
Exercises
"5
ENTER
¯
2nd
4"5
"80 5 "16"5 5 4"5 O
R P
12 cm 8 cm
Developing Skills
In 3 and 4,ABC is circumscribed about circle O and D, E, and F are points of tangency.
3. If AD 5, EB 5, and CF 10, find the lengths of the sides of the triangle and show that the triangle is isosceles.
4. If AF 10, CE 20, and BD 30, find the lengths of the sides of the triangle and show that the triangle is a right triangle.
In 5–11, is tangent to circle O at P, is tangent to circle O at S, and intersects circle O at T and R.
13. Line is tangent to circle O at P and intersects the circle at M, the midpoint of . If RP 3.00 cm, find the length of the radius of the circle:
a. in radical form b. to the nearest hundredth
14. Points E, F, G, and H are the points of tangency to circle O of , , , and , respectively. The measure of is 80°, of is 70°, and of is 50°. Find:
a. mEOF b. mFOG c. mGOH d. mHOE e. mAOE
f. mEAO g. mEAH h. mFBE i. mGCF j. mHDG
k. the sum of the measures of the angles of quadrilateral ABCD
GH
X
DA EFX
FGX
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E
Applying Skills
15. Prove Corollary 13.11a, “If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents.”
16. Prove Corollary 13.11b, “If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle whose vertex is the center of the circle and whose rays are the two radii drawn to the points of tangency.”
17. Lines and are tangent to circle O at Q and R.
a. Prove that PQR PRQ.
b. Draw intersecting at S and prove that QS RS and .
c. If OP 10, SQ 4 and OS SP, find OS and SP.
18. Tangents and to circle O are perpendicular to each other at C. Prove:
a.
b. OC
c. AOBC is a square.
19. Isosceles ABC is circumscribed about circle O. The points of tangency of the legs, and , are D and F, and the point of tangency of the base, , is E. Prove that E is the mid-point of .
20. Line is a common external tangent to circle O and circle O. is tangent to circle O at A
21. Line is a common internal tangent to circles O and O.
is tangent to circle O at A and to circle O at B, and
Hands-On Activity
Consider any regular polygon. Construct the angle bisectors of each interior angle. Since the interior angles are all congruent, the angles formed are all congruent. Since the sides of the regular polygon are all congruent, congruent isosceles triangles are formed by ASA. Any two adjacent triangles share a common leg. Therefore, they all share the same vertex. Since the legs of the triangles formed are all con-gruent, the vertex is equidistant from the vertices of the regular poly-gon. This common vertex is the centerof the regular polygon.
In this Hands-On Activity, we will use the center of a regular polygon to inscribe a circle in the polygon.
a. Using geometry software or compass, protractor, and straightedge, construct a square, a regular pentagon, and a regular hexagon. For each figure:
(1) Construct the center P of the regular polygon. (The center is the intersection of the angle bisectors of a regular polygon.)
(2) Construct an apothemor perpendicular from P to one of the sides of regular polygon.
(3) Construct a circle with center P and radius equal to the length of the apothem.
b. Prove that the circles constructed in part a are inscribed inside of the polygon. Prove:
(1) The apothems of each polygon are all congruent.
(2) The foot of each apothem is on the circle.
(3) The sides of the regular polygon are tangent to the circle.
c. Let r be the distance from the center to a vertex of the regular polygon. Since the center is equidistant from each vertex, it is possible to circumscribe a circle about the polygon with radius r. Let a be the length of an apothem and s be the length of a side of the regular polygon.
How is the radius, r, of the circumscribed circle related to the radius, a, of the inscribed circle?