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Splash Screen. Five-Minute Check (over Chapter 4) Then/Now New Vocabulary Theorems: Perpendicular Bisectors Example 1: Use the Perpendicular Bisector Theorems Theorem 5.3: Circumcenter Theorem Proof: Circumcenter Theorem Example 2: Real-World Example: Use the Circumcenter Theorem

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 4) Then/Now New Vocabulary Theorems: Perpendicular Bisectors Example 1: Use the Perpendicular Bisector Theorems Theorem 5.3: Circumcenter Theorem Proof: Circumcenter Theorem Example 2: Real-World Example: Use the Circumcenter Theorem Theorems: Angle Bisectors Example 3: Use the Angle Bisector Theorems Theorem 5.6: Incenter Theorem Example 4: Use the Incenter Theorem Lesson Menu

  3. A B C Classify the triangle. A. scalene B. isosceles C. equilateral 5-Minute Check 1

  4. A B C D Find x if mA = 10x + 15, mB = 8x – 18, andmC = 12x + 3. A. 3.75 B. 6 C. 12 D. 16.5 5-Minute Check 2

  5. A B C Name the corresponding congruent sides if ΔRST ΔUVW. A. R  V,S  W,T  U B. R  W,S  U,T  V C. R  U,S  V,T  W D. R  U,S  W,T  V 5-Minute Check 3

  6. A B C A. B. C. D. , Name the corresponding congruent sides if ΔLMN ΔOPQ. 5-Minute Check 4

  7. A B C D Find y if ΔDEF is an equilateral triangle and mF = 8y + 4. A. 22 B. 10.75 C. 7 D. 4.5 5-Minute Check 5

  8. A B C D ΔABC has vertices A(–5, 3) and B(4, 6). What are the coordinates for point C if ΔABC is an isosceles triangle with vertex angle A? A. (–3, –6) B. (4, 0) C. (–2, 11) D. (4, –3) 5-Minute Check 6

  9. You used segment and angle bisectors. (Lesson 1–3 and 1–4) • Identify and use perpendicular bisectors in triangles. • Identify and use angle bisectors in triangles. Then/Now

  10. perpendicular bisector • concurrent lines • point of concurrency • circumcenter • incenter Vocabulary

  11. Concept

  12. Use the Perpendicular Bisector Theorems A. Find the measure of BC. BC = AC Perpendicular Bisector Theorem BC = 8.5 Substitution Answer: 8.5 Example 1

  13. Use the Perpendicular Bisector Theorems B. Find the measure of XY. Answer: 6 Example 1

  14. Use the Perpendicular Bisector Theorems C. Find the measure of PQ. PQ = RQ Perpendicular Bisector Theorem 3x + 1 = 5x – 3 Substitution 1 = 2x – 3 Subtract 3x from each side. 4 = 2x Add 3 to each side. 2 = x Divide each side by 2. So, PQ = 3(2) + 1 = 7. Answer: 7 Example 1

  15. A B C D A. Find the measure of NO. A. 4.6 B. 9.2 C. 18.4 D. 36.8 Example 1

  16. A B C D B. Find the measure of TU. A. 2 B. 4 C. 8 D. 16 Example 1

  17. A B C D C. Find the measure of EH. A. 8 B. 12 C. 16 D. 20 Example 1

  18. Concept

  19. Concept

  20. Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points. Example 2

  21. Use the Circumcenter Theorem Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle. Example 2

  22. A B BILLIARDSA triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle? A. No, the circumcenter of an acute triangle is found in the exterior of the triangle. B. Yes, circumcenter of an acute triangle is found in the interior of the triangle. Example 2

  23. Concept

  24. Use the Angle Bisector Theorems A. Find DB. DB = DC Angle Bisector Theorem DB = 5 Substitution Answer:DB = 5 Example 3

  25. Use the Angle Bisector Theorems B. FindWYZ. Example 3

  26. Use the Angle Bisector Theorems WYZ  XYZ Definition of angle bisector mWYZ = mXYZ Definition of congruent angles mWYZ = 28 Substitution Answer:mWYZ = 28 Example 3

  27. Use the Angle Bisector Theorems C. Find QS. QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution x – 1 = 2 Subtract 3x from each side. x = 3 Add 1 to each side. Answer: So, QS = 4(3) – 1 or 11. Example 3

  28. A B C D A. Find the measure of SR. A. 22 B. 5.5 C. 11 D. 2.25 Example 3

  29. A B C D B. Find the measure of HFI. A. 28 B. 30 C. 15 D. 30 Example 3

  30. A B C D C. Find the measure of UV. A. 7 B. 14 C. 19 D. 25 Example 3

  31. Concept

  32. Use the Incenter Theorem A. Find SU if S is the incenter of ΔMNP. Find SU by using the Pythagorean Theorem. a2 + b2 = c2 Pythagorean Theorem 82 + SU2 = 102 Substitution 64 + SU2 = 100 82 = 64, 102 = 100 SU2 = 36 Subtract 64 from each side. SU = ±6 Take the square root of each side. Example 4

  33. Use the Incenter Theorem Since length cannot be negative, use only the positive square root, 6. Answer:SU = 6 Example 4

  34. Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, TNU = 2mSNU, so mTNU = 2(28) or 56. Use the Incenter Theorem B. FindSPU if S is the incenter of ΔMNP. Example 4

  35. Since SP bisects UPR, 2mSPU = UPR. This means that mSPU = UPR. 1 1 __ __ 2 2 Answer:mSPU = (62) or 31 Use the Incenter Theorem UPR + RMT + TNU = 180 Triangle Angle Sum Theorem UPR + 62 + 56 = 180 Substitution UPR + 118 = 180 Simplify. UPR = 62 Subtract 118 from each side. Example 4

  36. A B C D A. Find the measure of GF if D is the incenter of ΔACF. A. 12 B. 144 C. 8 D. 65 Example 4

  37. A B C D B. Find the measure of BCD if D is the incenter of ΔACF. A. 58° B. 116° C. 52° D. 26° Example 4

  38. End of the Lesson

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