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

Noah Atkinson · Accepted Answer

Endpoints of altitude are #(8,7)# and #(4,3)# and length of altitude is #4sqrt2#.

As the triangle has corners A, B, and C located at #(5,2)#, #(2,5)#, and #(8,7)#, respectively and the altitude goes through C#(8,7)#, it is obvious it is perpendicular to line joining #(5,2)#, #(2,5)#. The slope of line joining #(5,2)#, #(2,5)# is #(5-2)/(2-5)# or #-1# and hence slope of the required altitude is #1#. Now using point-slope form of equation, equation of altitude from #(8.7)# is #(y-7)=1*(x-8)# or #y-7=x-8# or #x-y=1# Further, equation of line joining AB is given by #(y-2)/(x-5)=-1# i.e. #y-2=-x+5# or #x+y=7# As equation of line joining AB is #x+y=7# and equation of altitude is #x-y=1#, their solution gives us base point of altitude which can be easily got as #(4,3)# (by adding and subtracting two equations) and distance of #(4,3)# from #(8,7)# is given by #sqrt((7-3)^2+(8-4)^2# or #sqrt32# or #4sqrt2#. Hence endpoints of altitude are #(8,7)# and #(4,3)# and length of altitude is #4sqrt2#.

Violet Aldrich · Answer

undefined To find the endpoints and length of the altitude going through corner C of the triangle with vertices A(5, 2), B(2, 5), and C(8, 7), follow these steps:

1. Determine the slope of the line passing through A and B using the formula:
$$ m_{AB} = \frac{y_B - y_A}{x_B - x_A} $$

2. The altitude passing through corner C is perpendicular to the line segment AB. Therefore, the slope of the altitude, $ m_{altitude} $, is the negative reciprocal of the slope of AB.

3. Calculate the negative reciprocal of the slope of AB to find the slope of the altitude.

4. Using the point-slope form of a line equation:
$$ y - y_1 = m(x - x_1) $$
where $ (x_1, y_1) $ is a point on the line and $ m $ is the slope, substitute the coordinates of corner C and the slope of the altitude to find the equation of the altitude.

5. Solve the equation obtained in step 4 simultaneously with the equation of the line containing the segment AB to find the intersection point. This point represents one endpoint of the altitude.

6. Use the distance formula to find the length of the altitude, which is the distance between the intersection point found in step 5 and corner C.

7. Repeat steps 1-6 for the line segment BC to find the other endpoint of the altitude and verify the length.

By following these steps, you can determine the endpoints and length of the altitude going through corner C of the triangle.