A triangle has corners A, B, and C located at $(5, 2)$ $(5 ,2 )$ , $(7, 9)$ $(7 ,9 )$ , and $(9, 8)$ $(9 ,8 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

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A triangle has corners A, B, and C located at $(5, 2)$ $(5 ,2 )$ , $(7, 9)$ $(7 ,9 )$ , and $(9, 8)$ $(9 ,8 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2016-10-23T21:06:50

The endpoints are $(9, 8) and (\frac{365}{53}, \frac{3192}{371})$ $(9,8) and (365/53, 3192/371)$

The distance is $\approx 2.2$ $~~ 2.2$

Explanation:

Use the point-slope form of the equation of a line to write the equation of the line through points A and B:

$y - 2 = \frac{2 - 9}{5 - 7} (x - 5)$ $y - 2 = (2 - 9)/(5 - 7)(x - 5)$

$y - 2 = \frac{- 7}{- 2} (x - 5)$ $y - 2 = (-7)/(-2)(x - 5)$

$y - 2 = \frac{7}{2} x - \frac{35}{2}$ $y - 2 = (7)/(2)x - 35/2$

$y = \frac{7}{2} x - \frac{31}{2}$ $y = (7)/(2)x - 31/2$ [1]

We need the above form and the standard form:

$2 y - 7 x + 31 = 0$ $2y - 7x + 31 = 0$ [2]

The slope of the altitude through point is the negative reciprocal of the slope in equation [1], $- \frac{2}{7}$ $-2/7$

Use the point-slope form of the equation of a line to find the equation of the altitude through point C:

$y - 8 = - \frac{2}{7} (x - 9)$ $y - 8 = -2/7(x - 9)$

$y - 8 = - \frac{2}{7} x + \frac{18}{7}$ $y - 8 = -2/7x + 18/7$

$y = - \frac{2}{7} x + \frac{74}{7}$ $y = -2/7x + 74/7$ [3]

To find the x coordinate of the other endpoint, subtract equation 3 from equation [1]

$y - y = \frac{7}{2} x + \frac{2}{7} x - \frac{31}{2} - \frac{74}{7}$ $y - y = (7)/(2)x + 2/7x - 31/2 - 74/7$

$0 = \frac{53}{14} x - \frac{365}{14}$ $0 = (53)/(14)x - 365/14$

$\frac{53}{14} x = \frac{365}{14}$ $(53)/(14)x = 365/14$

$x = \frac{365}{53}$ $x = 365/53$

To find the y coordinate of the other endpoint, substitute the above into equation [3]:

$y = - \frac{2}{7} (\frac{365}{53}) + \frac{74}{7}$ $y = -2/7(365/53) + 74/7$

$y = \frac{3192}{371}$ $y = 3192/371$

Use equation [2] to find the length of the altitude:

$d = \frac{| 2 (8) - 7 (9) + 31 |}{\sqrt{2^{2} + {(- 7)}^{2}}}$ $d = |2(8) - 7(9) + 31|/sqrt(2^2 + (-7)^2)$

$d \approx 2.2$ $d ~~ 2.2$

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A triangle has corners A, B, and C located at $(5, 2)$ $(5 ,2 )$ , $(7, 9)$ $(7 ,9 )$ , and $(9, 8)$ $(9 ,8 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer

Explanation:

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A triangle has corners A, B, and C located at (5,2)(5 ,2 ), (7,9)(7 ,9 ), and (9,8)(9 ,8 ), respectively. What are the endpoints and length of the altitude going through corner C?

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Explanation:

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A triangle has corners A, B, and C located at $(5, 2)$ $(5 ,2 )$ , $(7, 9)$ $(7 ,9 )$ , and $(9, 8)$ $(9 ,8 )$ , respectively. What are the endpoints and length of the altitude going through corner C?