A triangle has corners A, B, and C located at #(2 ,7 )#, #(5 ,3 )#, and #(2 , 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 #(2   ,7   )#, #(5  ,3  )#, and #(2 ,  4   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Sadie Adams · Accepted Answer

The end points of the altitude are #(2,4) and (86/25,127/25)#
The length is #9/5# Determine the line segment AB's equation and slope. #m_(AB) = (3 - 7)/(5-2)# #m_(AB) = -4/3# To determine the equation of the line segment AB, use one of the points and the slope: #y = -4/3(x - 2)+7" [1]"# To calculate the slope of the altitude, use the slope of AB. #m_("altitude") = -1/m_(AB)# #m_("altitude") = -1/(-4/3)# #m_("altitude") = 3/4# To determine the altitude equation, use point C and the slope: #y = 3/4(x - 2)+4" [2]"# To determine the x coordinate of the point of intersection, use equations [1] and [2]: #-4/3(x - 2)+7 = 3/4(x - 2)+4# #-4/3(x - 2)- 3/4(x - 2) = -3# #16(x - 2)+ 9(x - 2) = 36# #25(x - 2) = 36# #x - 2 = 36/25# #x = 50/25+36/25# #x = 86/25# To find the y coordinate, use any equation (I'll use [2]): #y = 3/4(36/25)+4# #y = 27/25+100/25# #y = 127/25# The end points of the altitude are #(2,4) and (86/25,127/25)# The separation between the two points is the altitude's length: #d = sqrt((86/25-2)^2+(127/25-4)^2# #d = sqrt(36^2+27^2)/25# #d = sqrt(2025)/25# #d = 9/5#

Sadie Adams · Answer

undefined The endpoints of the altitude going through corner C are (2, 4) and the intersection point of the line passing through points A and B with the line perpendicular to it passing through point C. The length of the altitude can be calculated using the distance formula between the two endpoints.