1. GARDENS Maggie wants to plant a circular flower bed within a triangular area set off by three pathways. Which point of concurrency related to triangles would she use for the center of the largest circle that would fit inside the triangle? The incenter is the center of a circle that intersects each side of the triangle at one point. For this reason, the incenter always lies in the interior of a triangle. So, Maggie should use the incenter to find the center of the circle inscribed in the triangle. incenter In, K is the centroid and DK = 16. Find each length. 3. CD Here, G is a midpoint of So, CG = GD = 9. 18 4. FG By the Centroid Theorem,. 2. KH 18 By the Centroid Theorem,. We know that DK + KH = DH. 8 esolutions Manual - Powered by Cognero Page 1
PROOF Write an indirect proof. 5. Given: Given: Step 1: Assume that x < 9. Step 2: Make a table with several possibilities for x, assuming x < 9. When x < 9, 5x + 7 < 52. Step 3: The assumption leads to the contradiction of the given information that. Therefore the assumption that x < 9 must be false, so the original conclusion that must be true. Find each measure. 6. TQR Triangle QRS and triangle QRT are congruent by SAS. Therefore, angle SQR is congruent to angle TQR is 43º. 43 7. XZ Given: Step 1: Assume that x < 9. Step 2: Make a table with several possibilities for x, assuming x < 9. By AAA, Solve for x.. By CPCTC, When x < 9, 5x + 7 < 52. Step 3: The assumption leads to the contradiction of the given information that. Therefore the assumption that x < 9 must be false, so the original conclusion that must be true. Substitute in XZ. 23 esolutions Manual - Powered by Cognero Page 2
8. GEOGRAPHY The distance from Tonopah to Round Mountain is equal to the distance from Tonopah to Warm Springs. The distance from Tonopah to Hawthorne is the same as the distance from Tonopah to Beatty. Determine which distance is greater, Round Mountain to Hawthorne or Warm Springs to Beatty. 9. MULTIPLE CHOICE If the measures of two sides of a triangle are 3.1 feet and 4.6 feet, which is the least possible whole number measure for the third side? A 1.6 feet B 2 feet C 7.5 feet D 8 feet Let n represent the length of the third side. According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides. Since we have two pairs of congruent distances, as described in the given information, and we know that 95 o >80 o, we can use the Hinge Theorem to prove that the distance from Warm Springs to Beatty is greater than the distance from Round Mountain to Hawthorne. Warm Springs to Beatty If n is the largest side, then n must be less than 3.1 + 4.6. Therefore, n < 7.7. If n is not the largest side, then 4.6 is the largest and 4.6 must be less than 3.1 + n. Therefore, 1.5 < n. Combining these two inequalities, we get 1.5 < n < 7.7. So, the least possible measure of the third side is the first whole number greater than 1.5, or 2. The correct option is B. B esolutions Manual - Powered by Cognero Page 3
Point H is the incenter of measure.. Find each 11. BD Since H is equidistant from the sides of, by the Incenter Theorem DH = FH. Find FH using the Pythagorean Theorem. 10. DH Since H is equidistant from the sides of, by the Incenter Theorem DH = FH. Find FH using the Pythagorean Theorem. Since length cannot be negative, use only the positive square root, 7. Since DH = FH, DH = 7. Use the Pythagorean Theorem in. Since length cannot be negative, use only the positive square root, 7. Since DH = FH, DH = 7. 7 Since length cannot be negative, use only the positive square root, 8.5. 8.5 esolutions Manual - Powered by Cognero Page 4
12. m HAC Here,. Also,. So by the Triangle Angle Sum Theorem,. Since is an angle bisector of angle A, 14. MULTIPLE CHOICE If the lengths of two sides of a triangle are 5 and 11, what is the range of possible lengths for the third side? F 6 < x < 10 G 5 < x < 11 H 6 < x < 16 J x < 5 or x > 11 Let n represent the length of the third side. According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides. Simplify. 32 13. m DHG Here,. Also,. If n is the largest side, then n must be less than 5 + 11. Therefore, n < 16. If n is not the largest side, then 11 is the largest and 11 must be less than 5 + n. Therefore, 6 < n. Combining these two inequalities, we get 6 < n < 16. So, the correct option is H. H Compare the given measures. 15. AB and BC 120 In and, we know that and. By the Hinge Theorem, since, we can state that AB < BC. AB < BC esolutions Manual - Powered by Cognero Page 5
16. RST and JKL Use the figure to determine which angle has the greatest measure. In and, we know that and. By the Converse of the Hinge Theorem, since, we know that. m RST > m JKL State the assumption necessary to start an indirect proof of each statement. 17. If 8 is a factor of n, then 4 is a factor of n. The first step in writing an indirect proof is to assume that the conclusion is false. 4 is not a factor of n. 4 is not a factor of n. 18. m M > m N The first step in writing an indirect proof is to assume that the conclusion is false. 19. If, then. The first step in writing an indirect proof is to assume that the conclusion is false. a > 7 20. 1, 5, 6 In the figure, is an obtuse angle, is an acute angle, and is also an acute angle. So, among these three angles has the greatest measure. 1 21. 9, 8, 3 In the figure, is an obtuse angle, is a right angle, and is an acute angle. So, among these three angles has the greatest measure. 8 22. 4, 3, 2 In the figure, is a right angle, is an acute angle, and is also an acute angle. So, among these three angles has the greatest measure. 4 a > 7 esolutions Manual - Powered by Cognero Page 6
PROOF Write a two-column proof. 23. Given: bisects SRT. m SQR > m SRQ Find the range for the measure of the third side of a triangle given the measures of the two sides. 24. 10 ft, 16 ft Let x represent the length of the third side. Next, set up and solve each of the three triangle inequalities. Given: bisects. Proof: Statements (Reasons) 1. bisects. (Given) 2. (Def. of bisector) 3. (Def. of angles) 4. (Exterior Angle Theorem) 5. (Def. of Inequality) 6. (Substitution) Given: bisects SRT. m SQR > m SRQ Notice that x > 6 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both inequalities is x > 6 and x < 26, which can be written as 6 ft < x < 26 ft. 6 ft < x < 26 ft 25. 23 m, 39 m Let x represent the length of the third side. Next, set up and solve each of the three triangle inequalities. Proof: Statements (Reasons) 1. bisects SRT. (Given) 2. (Def. of bisector) 3. m QRS = m QRT (Def. of s) 4. m SQR = m T + m QRT (Exterior Angle Theorem) 5. m SQR > m QRT (Def. of Inequality) 6. m SQR > m SRQ (Substitution) Notice that x > 16 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both inequalities is x > 16 and x < 62, which can be written as 16 m < x < 62 m. 16 m < x < 62 m esolutions Manual - Powered by Cognero Page 7