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

Oct 23, 2016

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

The distance is $\approx 2.2$

#### Explanation:

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

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

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

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

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

We need the above form and the standard form:

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

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

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

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

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

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

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

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

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

$\frac{53}{14} x = \frac{365}{14}$

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

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

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

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

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

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

$d \approx 2.2$