1. 5.4 Use Medians and Altitudes You will use medians and altitudes of triangles. Essential Question: How do you find the centroid of a triangle? You will learn how to answer this question by finding the point where the medians of a triangle intersect.

2. Warm-Up Exercises EXAMPLE 1 Use the centroid of a triangle SOLUTION SQ = 2 3 SW Concurrency of Medians of a Triangle Theorem 8 =8 = 2 3 SW Substitute 8 for SQ. 12 = SW Multiply each side by the reciprocal,. 3 2 Then QW = SW – SQ = 12 – 8 =4. So, QW = 4 and SW = 12. In RST, Q is the centroid and SQ = 8. Find QW and SW. Can the centroid and the incenter be the same point?

3. Warm-Up Exercises EXAMPLE 2 Standardized Test Practice SOLUTION Sketch FGH. Then use the Midpoint Formula to find the midpoint K of FH and sketch median GK. K( ) = 2 + 6, 5 + 1 22 K(4, 3) The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

4. Warm-Up Exercises EXAMPLE 2 Standardized Test Practice 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. 2 3 The coordinates of the centroid P are (4, 9 – 4), or (4, 5). The correct answer is B. How would you find the coordinates of the centroid using the median from F ?

5. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 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. 1. If SC = 2100 feet, find PS and PC. 700 ft, 1400 ftANSWER

6. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 2. If BT = 1000 feet, find TC and BC. 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. 1000 ft, 2000 ftANSWER

7. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 3. If PT = 800 feet, find PA and TA. 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. 1600 ft, 2400 ftANSWER

8. Warm-Up Exercises EXAMPLE 3 Find the orthocenter Find the orthocenter P in an acute, a right, and an obtuse triangle. SOLUTION Acute triangle P is inside triangle. Right triangle P is on triangle. Obtuse triangle P is outside triangle.

9. Warm-Up Exercises EXAMPLE 4 Prove a property of isosceles triangles Prove that the median to the base of an isosceles triangle is an altitude. SOLUTION GIVEN : ABC is isosceles, with base AC. BD is the median to base AC. PROVE : BD is an altitude of ABC.

10. Warm-Up Exercises EXAMPLE 4 Prove a property of isosceles triangles Proof : Legs AB and BC of isosceles ABC are congruent. CD AD because BD is the median to AC. Also, BD BD. Therefore, ABD CBD by the SSS Congruence Postulate.  ADB   CDB because corresponding parts of  s are . Also,  ADB and  CDB are a linear pair. BD and AC intersect to form a linear pair of congruent angles, so BD  AC and BD is an altitude of  ABC.

11. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 4. Copy the triangle in Example 4 and find its orthocenter. SOLUTION

12. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 5. WHAT IF? In Example 4, suppose you wanted to show that median BD is also an angle bisector. How would your proof be different? CBD ABD By SSS making ABD CBD which leads to BD being an angle bisector. ANSWER

13. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 6. Triangle PQR is an isosceles triangle and segment OQ is an altitude. What else do you know about OQ ? What are the coordinates of P ? OQ is also a perpendicular bisector, angle bisector, and median; (-h, 0). ANSWER

14. Warm-Up Exercises Daily Homework Quiz In Exercises 1–3, use the diagram. G is the centroid of ABC. 1. If BG = 9, find BF. ANSWER 13.5 2. If BD = 12, find AD. ANSWER 12 3. If CD = 27, find GC. ANSWER 18

15. Warm-Up Exercises Daily Homework Quiz ANSWER (1, 1) 5. Which type of triangle has Its orthocenter on the triangle. ANSWER a right triangle 4. Find the centroid of ABC.

16. You will use medians and altitudes of triangles. Essential Question: How do you find the centroid of a triangle? The medians of a triangle intersect at the centroid. Find the midpoints of the sides and draw the medians. Locate the point where the medians intersect.