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Splash Screen. Five-Minute Check (over Lesson 5–1) Then/Now New Vocabulary Theorem 5.7: Centroid Theorem Example 1: Use the Centroid Theorem Example 2: Use the Centroid Theorem Example 3: Real-World Example: Find the Centroid on a Coordinate Plane Key Concept: Orthocenter

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 5–1) Then/Now New Vocabulary Theorem 5.7: Centroid Theorem Example 1: Use the Centroid Theorem Example 2: Use the Centroid Theorem Example 3: Real-World Example: Find the Centroid on a Coordinate Plane Key Concept: Orthocenter Example 4: Find the Orthocenter on a Coordinate Plane Concept Summary: Special Segments and Points in Triangles Lesson Menu

  3. A B C D In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11. A. –5 B. 0.5 C. 5 D. 10 5-Minute Check 1

  4. A B C D In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13. A. 13 B. 11 C. 7 D. –13 5-Minute Check 2

  5. A B C D In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11). A. –12.5 B. 2.5 C. 10.25 D. 12.5 5-Minute Check 3

  6. A B C D In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ ___ A.DE B.DA C.DC D.DB ___ ___ ___ 5-Minute Check 4

  7. A B C D In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF? A. GCD B. DCG C. DFB D. ADE 5-Minute Check 5

  8. A B C D Which of the following statements about the circumcenter of a triangle is false? A. It is equidistant from the sides of the triangle. B. It can be located outside of the triangle. C. It is the point where the perpendicular bisectors intersect. D. It is the center of the circumscribed circle. 5-Minute Check 6

  9. You identified and used perpendicular and angle bisectors in triangles. (Lesson 5–1) • Identify and use medians in triangles. • Identify and use altitudes in triangles. Then/Now

  10. median • centroid • altitude • orthocenter Vocabulary

  11. Concept

  12. Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify. Example 1

  13. Use the Centroid Theorem YP + PV = YV Segment Addition 8 + PV = 12 YP = 8 PV = 4 Subtract 8 from each side. Answer:YP = 8; PV = 4 Example 1

  14. A B C D In ΔLNP, R is the centroid and LO = 30. Find LR and RO. A.LR = 15; RO = 15 B.LR = 20; RO = 10 C.LR = 17; RO = 13 D.LR = 18; RO = 12 Example 1

  15. Use the Centroid Theorem In ΔABC, CG = 4. Find GE. Example 2

  16. Use the Centroid Theorem Centroid Theorem Segment Addition and Substitution CG = 4 Distributive Property Example 2

  17. Subtract GE from each side. 1 __ 3 Use the Centroid Theorem Answer:GE = 2 Example 2

  18. A B C D In ΔJLN, JP = 16. Find PM. A. 4 B. 6 C. 16 D. 8 Example 2

  19. Find the Centroid on a Coordinate Plane SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance? Understand You need to find the centroid of the triangle. This is the point at which the triangle will balance. Example 3

  20. Find the Centroid on a Coordinate Plane Plan Graph and label the triangle with vertices (1, 4), (3, 0), and (3, 8). Use the Midpoint Theorem to find the midpoint of one of the sides of the triangle. The centroid is two-thirds the distance from the opposite vertex to that midpoint. Solve Graph ΔABC. Example 3

  21. Find the Centroid on a Coordinate Plane Find the midpoint D of side BC. Graph point D. Example 3

  22. Notice that is a horizontal line. The distance from D(3, 4 ) to A(1, 4) is 3 – 1 or 2 units. Find the Centroid on a Coordinate Plane Example 3

  23. The centroid is the distance. So, the centroid is (2) or units to the right of A. The coordinates are . Find the Centroid on a Coordinate Plane Example 3

  24. Answer: The artist should place the pole at the point Check Check the distance of the centroid from point D (3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid is . 1 2 __ __ 3 3 Find the Centroid on a Coordinate Plane Example 3

  25. A B C D 7 5 __ __ 3 3 A.(– , 2) B.(– , 2) C.(–1, 2) D.(0, 4) BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at (–3, 2), (–1, –2), and (–1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance? Example 3

  26. Concept

  27. Find the Orthocenter on a Coordinate Plane COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ. Example 4

  28. Find an equation of the altitude from The slope of so the slope of an altitude is Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Add 1 to each side. Example 4

  29. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6. Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Subtract 3 from each side. Example 4

  30. Substitution, Find the Orthocenter on a Coordinate Plane Then, solve a system of equations to find the point of intersection of the altitudes. Equation of altitude from J Multiply each side by 5. Add 105 to each side. Add 4x to each side. Divide each side by –26. Example 4

  31. Replace x with in one of the equations to find the y-coordinate. Rename as improper fractions. Multiply and simplify. Find the Orthocenter on a Coordinate Plane Example 4

  32. Answer:The coordinates of the orthocenter of ΔHIJ are Find the Orthocenter on a Coordinate Plane Example 4

  33. A B C D COORDINATE GEOMETRY The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC. A. (1, 0) B. (0, 1) C. (–1, 1) D. (0, 0) Example 4

  34. Concept

  35. End of the Lesson

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