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Warm Up

Use Medians and Altitudes. Warm Up. Lesson Presentation. Lesson Quiz. 1. For A (–4, 8) and B (5, 8), find the midpoint of AB. , 8. ANSWER. 1. 2. 2. For A (–3, 2) and B (4, –1), find the length of AB. 58. ANSWER. Warm-Up.

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Warm Up

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  1. Use Medians and Altitudes Warm Up Lesson Presentation Lesson Quiz

  2. 1.For A(–4, 8) and B(5, 8), find the midpoint of AB. , 8 ANSWER 1 2 2.For A(–3, 2) and B(4, –1), find the length of AB. 58 ANSWER Warm-Up

  3. 3.For A(0, 4) and C(18, 4), find the length of AB, where B is a point the distance from A to C. 2 3 12 ANSWER Warm-Up

  4. In RST, Qis the centroid and SQ = 8. Find QWand SW. SQ = SW 2 2 3 3 8= SW 3 2 Multiply each side by the reciprocal, . 12= SW 12 – 8 = 4. SW – SQ = Then QW = Example 1 SOLUTION Concurrency of Medians of a Triangle Theorem Substitute 8 for SQ. So, QW = 4 and SW = 12.

  5. Sketch FGH. Then use the Midpoint Formula to find the midpoint Kof FHand sketch median GK. 2 + 6 , 5 + 1 K( ) = K(4, 3) 2 2 Example 2 SOLUTION The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

  6. The distance from vertex G(4, 9)to K(4, 3)is 9–3 = 6 units. So, the centroid is (6) = 4 units down from G on GK. The correct answer is B. 2 3 Example 2 The coordinates of the centroid Pare (4, 9 – 4), or (4, 5).

  7. 1. 2. 3. If SC = 2100 feet, findPS andPC. If BT = 1000 feet, find TC andBC. If PT = 800 feet, findPA andTA. ANSWER 700 ft, 1400 ft ANSWER 1600 ft, 2400 ft ANSWER 1000 ft, 2000 ft Guided Practice There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P.

  8. Right triangle Pis on triangle. Acute triangle Pis inside triangle. Obtuse triangle P is outside triangle. Example 3 Find the orthocenter Pin an acute, a right, and an obtuse triangle. SOLUTION

  9. ABCis isosceles, with base AC. BDis the median to base AC. PROVE: BDis an altitude of ABC. Example 4 Prove that the median to the base of an isosceles triangle is an altitude. SOLUTION GIVEN:

  10. CD ADbecause BDis the median to AC. Also, BDBD. Therefore, ABD CBDby the SSS Congruence Postulate. ADB CDB because corresponding parts of s are . Also, ADB and CDBare a linear pair. BDand ACintersect to form a linear pair of congruent angles, so BD  ACand BDis an altitude of ABC. Example 4 Proof: Legs ABand BCof isosceles ABCare congruent.

  11. ANSWER Guided Practice 4. Copy the triangle in Example 4 and find its orthocenter.

  12. In Example 4, suppose you wanted to show that median BDis also an angle bisector. How would your proof be different? 5. WHAT IF? ANSWER ABD CBD by SSSmaking ABD CBD which leads to BD being an angle bisector. Guided Practice

  13. 6. Triangle PQRis an isoscleles triangle and OQis an altitude. What else do you know about OQ? What are the coordinates of P? ANSWER OQis also a perpendicular bisector, angle bisector, and median; (–h, 0). Guided Practice

  14. ANSWER ANSWER ANSWER 13.5 12 18 Lesson Quiz In Exercises 1–3, use the diagram. Gis the centroid of ABC. 1. If BG = 9, find BF. 2. If BD = 12, find AD. 3. If CD = 27, find GC.

  15. 4. Find the centroid of ABC. a right triangle ANSWER ANSWER (1, 1) Lesson Quiz 5. Which type of triangle has its orthocenter on the triangle?

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