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?

Question

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?

Noah Atkinson · Accepted Answer

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

The distance is #~~ 2.2# Write the equation of the line through points A and B using the point-slope form of the equation of a line: #y - 2 = (2 - 9)/(5 - 7)(x - 5)# #y - 2 = (-7)/(-2)(x - 5)# #y - 2 = (7)/(2)x - 35/2# #y = (7)/(2)x - 31/2# [1] Both the above form and the standard form are required: #2y - 7x + 31 = 0# [2] The slope of the altitude through point is the negative reciprocal of the slope in equation [1], #-2/7# To determine the equation of the altitude through point C, use the point-slope form of the equation of a line: #y - 8 = -2/7(x - 9)# #y - 8 = -2/7x + 18/7# #y = -2/7x + 74/7# [3] The other endpoint's x coordinate can be found by deducting equation 3 from equation [1]. #y - y = (7)/(2)x + 2/7x - 31/2 - 74/7# #0 = (53)/(14)x - 365/14# #(53)/(14)x = 365/14# #x = 365/53# Put the above into equation [3] to determine the other endpoint's y coordinate: #y = -2/7(365/53) + 74/7# #y = 3192/371# To determine the altitude's length, apply equation [2]: #d = |2(8) - 7(9) + 31|/sqrt(2^2 + (-7)^2)# #d ~~ 2.2#

Sadie Allen · Answer

undefined To find the endpoints of the altitude going through corner C, you first need to find the equation of the line containing side AB. Then, you'll find the midpoint of side AB, which will be one endpoint of the altitude. The other endpoint will be corner C itself. After that, you can calculate the length of the altitude by finding the distance between the endpoint at corner C and the line containing side AB.

1. Find the equation of the line containing side AB using the coordinates of points A and B.
$$ \text{Slope of AB} = \frac{y_B - y_A}{x_B - x_A} $$
$$ \text{Slope of AB} = \frac{9 - 2}{7 - 5} = \frac{7}{2} $$
$$ \text{Using point-slope form:} $$
$$ y - y_A = m(x - x_A) $$
$$ y - 2 = \frac{7}{2}(x - 5) $$
$$ 2y - 4 = 7x - 35 $$
$$ 7x - 2y = 31 $$

2. Find the midpoint of side AB, which will be one endpoint of the altitude.
$$ \text{Midpoint} = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) $$
$$ \text{Midpoint} = \left(\frac{5 + 7}{2}, \frac{2 + 9}{2}\right) $$
$$ \text{Midpoint} = (6, 5.5) $$

3. Use the midpoint and corner C to find the equation of the line perpendicular to side AB passing through corner C.
$$ \text{Slope of perpendicular line} = -\frac{1}{\text{Slope of AB}} = -\frac{2}{7} $$
$$ \text{Using point-slope form:} $$
$$ y - y_C = m(x - x_C) $$
$$ y - 8 = -\frac{2}{7}(x - 9) $$
$$ 7y - 56 = -2x + 18 $$
$$ 2x + 7y = 74 $$

4. Solve the system of equations formed by the two lines to find the intersection point, which is the endpoint of the altitude.
$$ \begin{cases} 7x - 2y = 31 \ 2x + 7y = 74 \end{cases} $$
$$ \text{Solve for } x \text{ and } y \text{ to get the endpoint of the altitude.} $$

5. Once you have the endpoints, use the distance formula to find the length of the altitude.