Name Date P R U. In Exercises 4 7, find the indicated measure. Explain your reasoning. D 4x + 5 C I
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1 ame ate 6.1 ractice In xercises 1 3, tell whether the information in the diagram allows you to conclude that point lies on the perpendicular bisector of, or on the angle bisector of. xplain your reasoning. 1. T. 3. T U In xercises 4 7, find the indicated measure. xplain your reasoning x + 5 x + 11 H I m 14 (3x + 8) (5x 1) 8. Write an equation of the perpendicular bisector of the segment with the endpoints, and 6, 0. ( ) ( ) 9. xplain how you can use the perpendicular bisector of a segment to draw an isosceles triangle. 10. In a right triangle, is it possible for the bisector of the right angle to be the same line as the perpendicular bisector of the hypotenuse? xplain your reasoning. raw a picture to support your answer. opyright ig Ideas earning, ll rights reserved. eometry esources by hapter 195
2 ame ate 6.1 ractice In xercises 1 3, tell whether the information in the diagram allows you to conclude that point lies on the perpendicular bisector of, or on the angle bisector of. xplain your reasoning In xercises 4 6, find the indicated measure. xplain your reasoning m 6. m UTW x + 5 3(x 8) 60 (0, 5) y 5 5 (, ) (x + 3) T (5x 4) 30 x (5, 0) U 9 V 9 W 7. Write an equation of the perpendicular bisector of the segment with the endpoints 3, 7 and H 1, 5. ( ) ( ) 8. In the figure, line m is the perpendicular 9. You are installing a fountain in the triangular bisector of. Is point on line m? Is garden pond shown in the point on line m? xplain your reasoning. figure. You want to place the fountain the same distance from each side of the pond. escribe a m way to determine the location of the fountain using angle 1 10 bisectors. 196 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
3 ame ate 6. ractice In xercises 1 3, the perpendicular bisectors of intersect at point, or the angle bisectors of XYZ intersect at point. ind the indicated measure. Tell which theorem you used X Y 3 T U Z In xercises 4 and 5, find the coordinates of the circumcenter of the triangle with the given vertices. 4. ( 6, 0 ), ( 0, 0 ), ( 0, 4) 5. U( 0, 0 ), V( 4, 0 ), W( 6, 6) In xercises 6 and 7, is the incenter of. Use the given information to find the indicated measure. 6. = 4x 8, = x = 6x +, = 8x 14 ind. ind. 8. raw an obtuse isosceles triangle. ind the circumcenter. Then construct the circumscribed circle. 9. cellular phone company is building a tower at an equal distance from three large apartment buildings. xplain how you can use the figure at the right to determine the location of the cell tower. uilding 1 uilding uilding Your friend says that it is impossible for the circumcenter of a triangle to lie outside the triangle. Is your friend correct? xplain your reasoning. 00 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
4 ame ate 6. ractice In xercises 1 3, find the indicated measure. Tell which theorem you used H 4 6 H ind the coordinates of the circumcenter of the triangle with the vertices 4, 1, 14, 6, and 6,. ( ) ( ) ( ) In xercises 5 and 6, use the diagram and the given information to find the indicated measures. 5. = 6x 14, = 3x + 6. = 4x, = 3x +, = x + 8 ind and. ind and. 7. You are using a rotary sprinkler to water the triangular lawn. a. xplain how to locate the sprinkler the same distance from each side of the triangular lawn. b. xplain how to locate the sprinkler the same distance from each vertex of the triangular lawn. c. Which is closer to vertex, the incenter or the circumcenter? xplain your reasoning. 8. xplain when the circumcenter of a triangle lies outside the triangle. 9. In the figure at the right, what value of x makes the incenter of? x 3 13 T 1 opyright ig Ideas earning, ll rights reserved. eometry esources by hapter 01
5 ame ate 6.3 ractice In xercises 1 3, point is the centroid of. Use the given information to find the indicated segment lengths. 1. = 1. = 7 3. = 10 ind and. ind and. ind and. 4. ind the coordinates of the centroid of the triangle with the vertices 6, 8, 3, 1, and 0, 3. ( ) ( ) ( ) In xercises 5 and 6, tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter. 5. ( 1, 5 ), ( 4, 3 ), ( 1, ) 6. ( 4, 6 ), ( 3, ), (, 6) 7. iven two vertices and the centroid of a triangle, how many possible locations are there for the third vertex? xplain your reasoning. 8. iven two vertices and the orthocenter of a triangle, how many possible locations are there for the third vertex? xplain your reasoning. 9. The centroid of a triangle is at (, 1) and vertices at ( 3, 5) and ( ) the third vertex of the triangle. 7, 4. ind 10. The orthocenter of a triangle is at the origin, and two of the vertices of the triangle 5, 0 3, 4. ind the third vertex of the triangle. are at ( ) and ( ) 11. Your friend claims that it is possible to draw an equilateral triangle for which the circumcenter, incenter, centroid, and orthocenter are not all the same point. o you agree? xplain your reasoning. 1. Your friend claims that when the median from one vertex of a triangle is the same as the altitude from the same vertex, the median divides the triangle into two congruent triangles. o you agree? xplain your reasoning. 13. an the circumcenter and the incenter of an obtuse triangle be the same point? xplain. 06 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
6 ame ate 6.4 ractice In xercises 1 5, use the graph of. y 1. In, show that the midsegment is parallel to and that = 1.. ind the coordinates of the endpoints of midsegment, which is opposite. 3. how that is parallel to and that = tate the coordinates of the endpoints of midsegment. 5. how that is parallel to and = 1. 1 x In xercises 6 11, use where,, and are the midpoints of the sides. 6. When = 16, what is? 7. When = 68, what is? 8. When = 46, what is? 9. When = 3x 1 and = 5x + 4, what is? 10. When = 6 x and = 5x 8, what is? 11. When = 5x + and = x + 5, what is? 1. Your friend claims that because each midsegment is half as long as the corresponding side of the triangle, the perimeter of the midsegment triangle is half the perimeter of the original triangle. Is your friend correct? xplain your reasoning. 13. building has the shape of a pyramid with a square base. The midsegment parallel to the ground of each triangular face of the pyramid has a length of 58 feet. ind the length of the base the pyramid. 58 ft 10 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
7 ame ate 6.4 ractice In xercises 1 4, use the graph of. y 1. ind the coordinates of the midpoint of, the midpoint of, and the midpoint of.. raph the midsegment triangle,. 5 1 x 3. how that,, and how that = 1, = 1, and = 1. 5 In xercises 5 8, use. where U, V, and W are the midpoints of the sides. 5. When V = 9, what is UW? U = x 5 and VW = 8 x, what is? 6. When ( ) U V = x 1 + x and UW = x + 4, what is V? 7. When ( ) ( ) 8. When UV = y + 14 and = 13 y, what is W? W 9. The bottom two steps of a stairwell are shown. xplain how to use the given measures to verify that the bottom step is parallel to the floor. 13 in. 8 in. 13 in. 8 in. floor 10. Your friend claims that a triangle with side lengths of a, b, and c will have half the area of a triangle with side lengths of a, b, and c. Is your friend correct? xplain your reasoning. opyright ig Ideas earning, ll rights reserved. eometry esources by hapter 11
8 ame ate 6.5 ractice In xercises 1 and, list the angles of the given triangle from smallest to largest x x 5x 18 In xercises 3 and 4, list the sides of the given triangle from shortest to longest x 1 x (x 4) In xercises 5 and 6, is it possible to construct a triangle with the given side lengths? xplain , 37, , 16, 8 7. Write an indirect proof that a triangle has at most one obtuse angle. 8. escribe the possible values of x in 9. ist the angles of the given triangle the figure shown. from smallest to largest. xplain your reasoning. x + 1 x + 31 (6x + 4) x + 16 (9x + 16) (4x ) H 10. The shortest distance between two points is a straight line. xplain this statement in terms of the Triangle Inequality Theorem (Theorem 6.11). opyright ig Ideas earning, ll rights reserved. eometry esources by hapter 15
9 ame ate 6.5 ractice In xercises 1 and, list the angles of the given triangle from smallest to largest. 1.. U x + 1 V x + 8 x + 10 x 3 x x W In xercises 3 and 4, list the sides of the given triangle from shortest to longest Write an indirect proof that a right triangle has exactly two acute angles. 6. Is it possible to construct a triangle with side lengths ( x ) x + 41 if x = 9? xplain. 5 6, 3x + 80, and 7. The figure shows several triangles, with labeled side lengths. Which of the triangles are labeled correctly? xplain Your friend claims that if you are given the three angle measures of a triangle, you can construct a triangle that obeys the Triangle Inequality Theorem (Theorem 6.11), even if you are not given any of the side lengths. Is your friend correct? xplain your reasoning. 16 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
10 ame ate 6.6 ractice In xercises 1 4, copy and complete the statement with <, >, or =. xplain your reasoning. 1.. m HI m I H 6 I 6 3. m 1 m In xercises 5 and 6, write and solve an inequality for the possible values of x. 5. (x + 7) 6. 3 (x 3) In xercises 7 and 8, write a proof. 6(x + 1) 14x iven: TV UW, TU > VW 8. iven: m 1 > m, is the midpoint of. rove: m TVU > m WUV rove: > T U 1 V W 9. The figure shows two sliding boards. The slide is the same length in each case, but one is steeper than the other. an you apply the Hinge Theorem (Theorem 6.1) or the onverse of the Hinge Theorem (Theorem 6.13) in this problem? xplain your reasoning. 0 eometry opyright ig Ideas earning, esources by hapter ll rights reserved.
11 ame ate 6.6 ractice In xercises 1 4, copy and complete the statement with <, >, or =. xplain your reasoning. 1.. I H I H 3. m 1 m 4. m U m 1 3x T 4 5x 4 U In xercises 5 and 6, write and solve an inequality for the possible values of x. 5. (3x 8) x x (x + ) 7. Use the figure to write a proof. iven: rove: m > m 8. Two sailboats started at the same location. ailboat traveled 5 miles west, then turned 9 toward the north and continued for 8 miles. ailboat first went south for 8 miles, then turned 51 toward the east and continued for 5 miles. Which sailboat was farther from the starting point? xplain your reasoning. 9. How are the Hinge Theorem (Theorem 6.1) and the ongruence Theorem (Theorem 5.5) similar? How are they different? xplain your reasoning. opyright ig Ideas earning, ll rights reserved. eometry esources by hapter 1
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