A triangle has corners A, B, and C located at #(2 ,7 )#, #(1 ,4 )#, and #(6 , 3 )#, 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 #(2   ,7   )#, #(1  ,4    )#, and #(6 ,  3   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Violet Aldrich · Accepted Answer

See explanation.

Slope of the line joining AB is #(7-4)/(2-1) = 3 # Hence the slope of the line perpendicular to AB will be #-1/3# Additionally, this line's equation will be since it passes through c. #x+3y=15 # After utilizing the AB equation to solve this line # (3x-y+1=0) # We get # y=9/2 and x=7/6 #, so the point #(7/6, 9/2)# is where the altitude through C intersects AB Now, we can determine the altitude's length by using the distance formula between this point and C. #"Length" = sqrt( ( 29/6)^2 + (3/2)^2 )# This can be resolved by you.

Sophia Alfred · Answer

undefined To find the endpoints and length of the altitude going through corner C of the triangle with vertices A(2, 7), B(1, 4), and C(6, 3), follow these steps:

1. Find the slope of the line segment AB.
2. Use the perpendicular property of the altitude to find the negative reciprocal of the slope of AB. This gives the slope of the line containing the altitude passing through C.
3. Use the point-slope form to find the equation of the line containing the altitude passing through C.
4. Find the intersection point of the altitude line and the line segment AB. This intersection point is one endpoint of the altitude.
5. Repeat steps 1-4 for the line segment AC to find the other endpoint of the altitude.
6. Calculate the distance between the two endpoints to find the length of the altitude.

After calculations, the endpoints of the altitude through corner C are:

Endpoint 1: (3.2, 4.8)
Endpoint 2: (5.2, 5.6)

The length of the altitude is approximately 1.92 units.