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Understanding Medians and Altitudes of Triangles: Centroid and Orthocenter Theorems

Learn how to apply the Centroid Theorem and identify the orthocenter of triangles. Examples and real-world scenarios guide you through finding key triangle points on a coordinate plane.

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Understanding Medians and Altitudes of Triangles: Centroid and Orthocenter Theorems

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  1. LESSON 5–2 Medians and Altitudes of Triangles

  2. Five-Minute Check (over Lesson 5–1) TEKS 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. 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. 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. 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. 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. 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. 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. Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(F), G.1(G) TEKS

  10. You identified and used perpendicular and angle bisectors in triangles. • Identify and use medians in triangles. • Identify and use altitudes in triangles. Then/Now

  11. median • centroid • altitude • orthocenter Vocabulary

  12. Concept

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

  14. 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

  15. 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

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

  17. Use the Centroid Theorem Centroid Theorem CG = 4 6 = CE Example 2

  18. Use the Centroid Theorem CG + GE = CE Segment Addition 4 + GE = 6 Substitution GE = 2 Subtract 4 from each side. Answer:GE = 2 Example 2

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

  20. 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? Analyze You need to find the centroid of the triangle. This is the point at which the triangle will balance. Example 3

  21. Find the Centroid on a Coordinate Plane Formulate Graph and label the triangle with vertices at (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. Determine Graph the triangle and label the vertices A, B, and C. Example 3

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

  23. 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

  24. The centroid P 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 P Example 3

  25. Answer: The artist should place the pole at the point Justify 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

  26. Find the Centroid on a Coordinate Plane Evaluate: Any side could have been used to find the centroid. Using all three sides is a simple way to check the solution. Point P appears to be in the middle of the triangle, so the answer is reasonable. Example 3

  27. A.B. 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

  28. Concept

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

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

  35. 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

  36. Concept

  37. LESSON 5–2 Medians and Altitudes of Triangles

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