that we draw auxiliary lines through N parallel to , through M parallel to that we show that the altitudes of are perpendicular bisectors of the

The following relationships are also implied by Theorem 7.2.4. In Figure 7.21, , , and . Equivalently, , , and . PC = 1 2 CT CN = 1 2 SC CM = 1 2 RC CT = 2PC SC = 2CN RC = 2 CM Solution Median separates into two congruent right triangles, and ; this follows from SSS. With Z the midpoint of , . Using the Pythagorean Theorem with in Figure 7.22b, we have By Theorem 7.2.4, Because it follows that QZ = 4. QZ = 1 2 RQ, RQ = 2 3 RZ = 2 3 12 = 8 RZ = 12 RZ 2 = 144 225 = RZ 2 + 81 15 2 = RZ 2 + 9 2 RS 2 = RZ 2 + SZ 2 䉭RZS SZ = 9 ST 䉭RZT 䉭RZS 䉭RST RZ 쮿 EXAMPLE 7 GIVEN: In Figure 7.22a, isosceles with , and ; medians , , and meet at centroid Q. FIND: RQ and QZ SY TX RZ ST = 18 RS = RT = 15 䉭RST R S T Y Q X Z a Figure 7.22 R S Q Z b 15 9 Exs. 17–22 It is possible for the angle bisectors of certain quadrilaterals to be concurrent. Likewise, the perpendicular bisectors of the sides of a quadrilateral can be concurrent. Of course, there are four angle bisectors and four perpendicular bisectors of sides to consider. In Example 8, we explore this situation. 쮿 Figure 7.23 EXAMPLE 8 Use intuition and Figure 7.23 to decide which of the following are concurrent. B D C A W Z X Y a The angle bisectors of a kite c The angle bisectors of a rectangle b The perpendicular bisectors of d The perpendicular bisectors of the the sides of a kite sides of a rectangle Solution a The angle bisectors of the kite are concurrent at a point the incenter of the kite. b The bisectors of the sides of the kite are not concurrent unless and are both right angles. c The angle bisectors of the rectangle are not concurrent unless the rectangle is a square. d The bisectors of the sides of the rectangle are concurrent the circumcenter of the rectangle is also the point of intersection of diagonals. NOTE: The student should make drawings to verify the results in Example 8. ⬜ ∠C ∠A ⬜ The centroid of a triangular region is sometimes called its center of mass or center of gravity . This is because the region of uniform thickness “balances” upon the point known as its centroid. Consider the Discover activity at the left. Discover Take a piece of cardboard or heavy poster paper. Draw a triangle on the paper and cut out the triangular shape. Now use a ruler to mark the midpoints of each side and draw the medians to locate the centroid. Place the triangle on the point of a pen or pencil at the centroid and see how well you can balance the triangular region. Exercises 7.2

1. In the figure, are lines m, n, and

p concurrent?

2. If one exists, name the point of

concurrence for lines m, n, and p.

5. What is the general name of the point of concurrence for

the three perpendicular bisectors of sides of a triangle?

6. What is the general name of the point of concurrence for

the three medians of a triangle?

7. Which lines or line segments or rays must be drawn or

constructed in a triangle to locate its a incenter? b circumcenter? c orthocenter? d centroid?

8. Is it really necessary to construct all three angle bisectors

of the angles of a triangle to locate its incenter?

9. Is it really necessary to construct all three perpendicular

bisectors of the sides of a triangle to locate its circumcenter?

10. To locate the orthocenter, is it necessary to construct all

three altitudes of a right triangle? A m n p Exercises 1, 2

3. What is the general name of the point of concurrence for

the three angle bisectors of a triangle?

4. What is the general name of the point of concurrence for

the three altitudes of a triangle?

11. For what type of triangle are the angle bisectors, the

medians, the perpendicular bisectors of sides, and the altitudes all the same?

12. What point on a right triangle is the orthocenter of the

right triangle?

13. What point on a right triangle is the circumcenter of the

right triangle?

14. Must the centroid of an isosceles triangle lie on the

altitude to the base? 15. Draw a triangle and, by construction, find its incenter. 16. Draw an acute triangle and, by construction, find its circumcenter.

17. Draw an obtuse triangle and, by construction, find its

circumcenter.

18. Draw an acute triangle and, by construction, find its

orthocenter.

19. Draw an obtuse triangle and, by construction, find its

orthocenter. HINT: You will have to extend the sides opposite the acute angles.

20. Draw an acute triangle and, by construction, find the

centroid of the triangle. HINT: Begin by constructing the perpendicular bisectors of the sides.

21. Draw an obtuse triangle and, by construction, find the

centroid of the triangle. HINT: Begin by constructing the perpendicular bisectors of the sides.

22. Is the incenter always located in the interior of the

triangle?

23. Is the circumcenter always located in the interior of the

triangle?

24. Find the length of the radius of the inscribed circle for a

right triangle whose legs measure 6 and 8.

25. Find the distance from the circumcenter to each vertex of

an equilateral triangle whose sides have the length 10.

26. A triangle has angles measuring 30°, 30°, and 120°. If the

congruent sides measure 6 units each, find the length of the radius of the circumscribed circle.

27. Given: Isosceles

and Medians , , and meet at centroid Q Find: RQ and SQ

28. Given: Isosceles

and Medians , , and meet at Q Find: RQ and QT SY TX RZ ST = 16 RS = RT = 10 䉭RST SY TX RZ ST = 16 RS = RT = 17 䉭RST

29. In , medians

, , and intersect at centroid Q. a If , find QB . b If , find PQ . c If , find AN . AQ = 3.5 QC = 3 MQ = 8 PC NA MB 䉭MNP R S T Q Y X Z Exercises 27, 28 M Q P N B C A Exercises 29, 30

30. In , medians

, , and intersect at centroid Q. a Find QB if . b Find PQ if . c Find AN if . 31. Draw a triangle. Construct its inscribed circle. 32. Draw a triangle. Construct its circumscribed circle.

33. For what type of triangle will the incenter and the

circumcenter be the same? 34. Does a rectangle have a an incenter? b a circumcenter? 35. Does a square have a an incenter? b a circumcenter? 36. Does a regular pentagon have a an incenter? b a circumcenter? 37. Does a rhombus have a an incenter? b a circumcenter? 38. Does an isosceles trapezoid have a an incenter? b a circumcenter?

39. A distributing company plans an

Illinois location that would be the same distance from each of its principal delivery sites at Chicago, St. Louis, and Indianapolis. Use a construction method to locate the approximate position of the distributing company. NOTE: Trace the outline of the two states on your own paper.

40. There are plans to locate a disaster

response agency in an area that is prone to tornadic activity. The agency is to be located at equal distances from Wichita, Tulsa, and Oklahoma City. Use a construction method to locate the approximate position of the agency. NOTE: Trace the outline of the two states on your own paper.

41. A circle is inscribed in an isosceles triangle with legs of

length 10 in. and a base of length 12 in. Find the length of the radius for the circle. AQ = 4.6 QC =

7 2

MQ = 8.2 PC NA MB 䉭MNP C SL I OKC T W Regular Polygon Center and Central Angle of a Regular Polygon Radius and Apothem of a Regular Polygon More About Regular Polygons 7.3 KEY CONCEPTS Several interesting properties of regular polygons are developed in this section. For in- stance, every regular polygon has both an inscribed circle and a circumscribed circle; fur- thermore, these two circles are concentric. In Example 1, we use bisectors of the angles of a square to locate the center of the inscribed circle. The center, which is found by using the bisectors of any two consecutive angles, is equidistant from the sides of the square. Reminder A regular polygon is both equilateral and equiangular. EXAMPLE 1 Given square ABCD in Figure 7.24a, construct inscribed . } O A D B C a Figure 7.24 A D B C b O c A M D B C O Solution Figure 7.24b: The center of an inscribed circle must lie at the same distance from each side. Center O is the point of concurrency of the angle bisectors of the square. Thus, we construct the angle bisectors of and to identify point O. Figure 7.24c: Constructing , OM is the distance from O to and the length of the radius of the inscribed circle. Finally we construct inscribed with radius as shown. OM } O AB OM ⬜ AB ∠C ∠B 쮿 In Example 2, we use the perpendicular bisectors of two consecutive sides of a reg- ular hexagon to locate the center of the circumscribed circle. The center determines a point that is equidistant from the vertices of the hexagon. EXAMPLE 2 Given regular hexagon MNPQRS in Figure 7.25a, construct circumscribed . } X R Q P S M N a Figure 7.25 R Q P X S M N b R Q P X S M N c