A triangle has corners A, B, and C located at #(3 ,8 )#, #(7 ,5 )#, and #(2 ,9 )#, 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 #(3   ,8   )#, #(7  ,5    )#, and #(2 ,9   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Aurora Ayala · Accepted Answer

Coordinates of CF (2,9) , (47/25, 221/25))

Length of Altitude CF = #color(purple)(0.2)# Equation of side AB is #(y-y1)/(y2-y1)=(x-x1)/(x2-x1)# #(y-8)/(5-8)=(x-3)/(7-3) # #4y - 32 = -3x + 9# #color(brown)(3x + 4y = 41, (Equation 1))# Slope of AB m1#=(y2-y1)/(x2-x1)=(5-8)/(7-3)=-(3/4)# Slope of altitude CF #m2=-1/(m1) = -1/(-3/4)= 4/3# Equation of altitude CF #(y-y3)=m2(x-x3)# #(y-9) = (4/3)(x-2)# #(3y - 27 = 4x - 8# #color(brown)(3y - 4x = 19, (Equation 2))# By solving #color(brown)(Equations 1 & 2)#, we get point F #x = 47/25, y = 221/25# Coordinates of #F (47/25, 221/25)# length of the CF altitude #CF=sqrt(((47/25)-2)^2+((221/25)-9)^2)# #CF=sqrt((-3/25)^2+(-4/25)^2)= color(brown)(1/5 = 0,2)#

Ella Armstrong · Answer

undefined The altitude going through corner C of the triangle with vertices at (3, 8), (7, 5), and (2, 9) has endpoints at (2, 9) and the foot of the altitude, which we'll denote as point D. To find the foot of the altitude, we need to find the intersection point of the line passing through A and B (the base of the triangle) and the perpendicular line passing through C.

1. Find the slope of line AB using the coordinates of points A and B.
2. Find the negative reciprocal of the slope of AB to get the slope of the line perpendicular to AB passing through point C.
3. Use the slope of the perpendicular line and the coordinates of point C to find the equation of the perpendicular line.
4. Find the intersection point of the perpendicular line and the line AB to get the coordinates of point D.

Once you have the coordinates of point D, you'll have the endpoints of the altitude. Then, you can calculate the length of the altitude using the distance formula between points C and D.