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Triangle Centroid and Side Length Calculation

Calculate the side lengths QW and SW of a triangle given the length of SQ as 8. Use the centroid theorem and reciprocal method to find the answers.

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Triangle Centroid and Side Length Calculation

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  1. In RST, Qis the centroid and SQ = 8. Find QWand SW. SQ = SW 2 2 3 3 8= SW 2 3 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.

  2. 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.

  3. 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).

  4. 1. If SC = 2100 feet, findPS andPC. PC SC = Concurrency of Medians 3 3 of a Triangle Theorem 2 2 PC 2100= 3 2 Multiply each side by the reciprocal, . 1400= PC 2100 –1400 = 700. SC – PC = Then PS = 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. SOLUTION Substitute 2100 for SC. So, PS = 700 and PC = 1400.

  5. 2. If BT = 1000 feet, find TC andBC. TC BT = 1000= TC for Examples 1 and 2 GUIDED PRACTICE SOLUTION T is midpoint of side BC. Substitute 1000 for BT. BC = 2 TC BC = 2000 So, BC = 2000 and TC = 1000.

  6. 3. If PT = 800 feet, findPA andTA. PT = PA Concurrency of Medians of 1 1 a Triangle Theorem 2 2 800= PA 1 2 Multiply each side by the reciprocal, . 1600= PA for Examples 1 and 2 GUIDED PRACTICE SOLUTION Substitute 800 for PT. 1600 + 800 = 2400. PT + PA = Then TA = So, TA = 2400 and PA = 1600.

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