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
Douglas K.
Oct 23, 2016

The endpoints are (9,8) and (365/53, 3192/371)(9,8)and(36553,3192371)

The distance is ~~ 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 = (2 - 9)/(5 - 7)(x - 5)y−2=2−95−7(x−5)

y - 2 = (-7)/(-2)(x - 5)y−2=−7−2(x−5)

y - 2 = (7)/(2)x - 35/2y−2=72x−352

y = (7)/(2)x - 31/2y=72x−312 [1]

We need the above form and the standard form:

2y - 7x + 31 = 02y−7x+31=0 [2]

The slope of the altitude through point is the negative reciprocal of the slope in equation [1], -2/7−27

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

y - 8 = -2/7(x - 9)y−8=−27(x−9)

y - 8 = -2/7x + 18/7y−8=−27x+187

y = -2/7x + 74/7y=−27x+747 [3]

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

y - y = (7)/(2)x + 2/7x - 31/2 - 74/7y−y=72x+27x−312−747

0 = (53)/(14)x - 365/140=5314x−36514

(53)/(14)x = 365/145314x=36514

x = 365/53x=36553

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

y = -2/7(365/53) + 74/7y=−27(36553)+747

y = 3192/371y=3192371

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

d = |2(8) - 7(9) + 31|/sqrt(2^2 + (-7)^2)d=|2(8)−7(9)+31|√22+(−7)2

d ~~ 2.2d≈2.2

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