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## EXAMPLE 1

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**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 Use the centroid of a triangle SOLUTION Concurrency of Medians of a Triangle Theorem Substitute 8 for SQ. So, QW = 4 and SW = 12.**Sketch FGH. Then use the Midpoint Formula to find the**midpoint Kof FHand sketch median GK. 2 + 6 , 5 + 1 K( ) = 2 2 EXAMPLE 2 Standardized Test Practice SOLUTION K(4, 3) The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.**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 Standardized Test Practice The coordinates of the centroid Pare (4, 9 – 4), or (4, 5).**1.**If SC = 2100 feet, findPS andPC. ANSWER 700 ft, 1400 ft for Examples 1 and 2 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.**2.**If BT = 1000 feet, find TC andBC. ANSWER 1000 ft, 2000 ft for Examples 1 and 2 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.**3.**If PT = 800 feet, findPA andTA. ANSWER 1600 ft, 2400 ft for Examples 1 and 2 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.