Explanation Given information : The triangle has vertices ( − 2 , 1 ) , ( 5 , 6 ) a n d ( 3 , − 2 ) .Three lines representing altitude are y A D = − 1 4 x + 1 2 , y B E = 5 3 x − 7 3 and y C F = − 7 5 x + 11 5 Formula used: The formula to find the midpoint (M) of two points ( x 1 , y 1 ) & ( x 2 , y 2 ) is M = ( x 1 + x 2 2 , y 1 + y 2 2 ) Concept Involved: The midpoints R, P and Q of sides AB, BC, & CA can be found using the midpoint formula. Set up the equation of altitude and solve them to find the point of intersection between each altitude. Calculation: Table 1: Finding the midpoints P, Q, & R Description Steps Label each vertices of the triangle A ( − 2 , 1 ) , B ( 5 , 6 ) a n d C ( 3 , − 2 ) Finding the midpoint of side AB P = ( 5 + 3 2 , 6 + − 2 2 ) = ( 8 2 , 4 2 ) = ( 4 , 2 ) Finding the midpoint of side BC Q = ( − 2 + 3 2 , 1 + − 2 2 ) = ( 1 2 , − 1 2 ) Finding the midpoint of side CA R = ( − 2 + 5 2 , 1 + 6 2 ) = ( 3 2 , 7 2 ) Table 2: Finding the slope of the medianAP, BQ, &CR Description Steps The slope of median drawn from point P ( 4 , 2 ) on BC to opposite vertices A ( − 2 , 1 ) m A P = 2 − 1 4 + 2 = 1 6 The slope of median drawn from point Q ( 1 2 , − 1 2 ) on CA to opposite vertices B ( 5 , 6 ) m B Q = 6 + 1 2 5 − 1 2 = 12 2 + 1 2 10 2 − 1 2 = 13 2 9 2 = 13 2 × 2 9 = 13 9 The slope of median drawn from pointR ( 3 2 , 7 2 ) on AB to opposite vertices C ( 3 , − 2 ) m C R = − 2 − 7 2 3 − 3 2 = − 4 2 − 7 2 6 2 − 3 2 = − 11 2 3 2 = − 11 2 × 2 3 = − 11 3 Table 3: Finding the equation of the medianAP, BQ, &CR Description Steps The equation of altitude AP with slope m A P = 1 6 passing through the point A ( − 2 , 1 ) y − 1 = 1 6 ( x + 2 ) y − 1 = 1 6 x + 2 6 y − 1 = 1 6 x + 1 3 y − 1 + 1 = 1 6 x + 1 3 + 1 y = 1 6 x + 1 3 + 3 3 y A P = 1 6 x + 4 3 The equation of altitude BQ with slope m B Q = 13 9 passing through the point B ( 5 , 6 ) y − 6 = 13 9 ( x − 5 ) y − 6 = 13 9 x + 13 ( − 5 ) 9 y − 6 = 13 9 x − 65 9 y − 6 + 6 = 13 9 x − 65 9 + 6 y = 13 9 x − 65 9 +<

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 1, Problem 55SP

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Toshow : the three lines (altitudes) intersect at the same point and find the coordinate of that point.

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Chapter 1, Problem 54SP

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Chapter 1 Solutions

Calculus

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Toshow : the three lines (altitudes) intersect at the same point and find the coordinate of that point.

Solution Summary: The author explains the formula to find the midpoints R, P, and Q of sides AB, BC, & CA.

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Toshow : the three lines (altitudes) intersect at the same point and find the coordinate of that point.

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Chapter 1 Solutions