A triangle has corners A, B, and C located at #(7 ,3 )#, #(4 ,8 )#, and #(3 , 4 )#, 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 #(7   ,3   )#, #(4  ,8    )#, and #(3 ,  4   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Wyatt Anderson · Accepted Answer

The answer is (5.5,5.5)

this an isosceles triangle, you can show it by calculating the #AC = BC # distance formula #sqrt((x_2-x_1)^2 + (y_2-y_1)^2)# Thus: # sqrt(4^2+1^2) = 4.123# #sqrt (1^2+4^2) = 4.123# So the midpoint of the base should be the altitude since it will also be the perpendicular bisector. Now subtracting point A from B You will find the horizontal a d vertical separation to be (3,-5) half these separation, (1.5, -2.5), add to B(4,8) or subtract fromA(7, 3) And get the point (5.5,5.5) this is the altitude of our isosceles triangle...

Evelyn Arboleda · Answer

undefined To find the endpoints of the altitude going through corner C of the triangle ABC, we first need to find the equation of the line containing side AB. Then, we'll find the midpoint of AB, which will be one endpoint of the altitude. The other endpoint will be corner C itself. Finally, we'll calculate the length of this altitude.

1. Find the equation of the line containing side AB:
   - Using the coordinates of points A and B, calculate the slope of AB using the formula:
     $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
   - Then, use the slope-intercept form of a line equation, $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. Substitute one of the points, say A, into the equation to find $ b $.

2. Find the midpoint of AB:
   - Use the midpoint formula:
     $$ M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$

3. The altitude through corner C will be perpendicular to side AB, so its slope will be the negative reciprocal of the slope of AB.

4. Use the point-slope form of a line equation, $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is a point on the line (the midpoint of AB).

5. Substitute the coordinates of point C into the equation to find the equation of the altitude line.

6. Finally, calculate the length of the altitude by finding the distance between the endpoints using the distance formula:
   $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Performing these steps will yield the endpoints and length of the altitude going through corner C.