A triangle has corners A, B, and C located at #(4 ,5 )#, #(3 ,6 )#, and #(2 ,9 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
Jan 28, 2017

The two end points of altitude are #(2,9) **and** (1,8)#
Length of altitude is #sqrt2#

Explanation:

CD is the altitude to be drawn from corner on line AB. Lines CD and AB are perpendicular to each other . Slope of the line AB is # m_1=(y_2-y_1)/(x_2-x_1) = (6-5)/(3-4) = -1#. Slope of the line CD is #m_2= (-1)/m_1=1#.Since product of slopes of perpendicular lines is #-1#.

Equation of line AB is #y-y_1=m_1(x-x_1) or y-5 = -1(x-4) or y+x=9 (1)#
Equation of line CD is #y-y_1=m_2(x-x_1) or y-9 = 1(x-2) or y-x=7 (2)#

Solving equation (1) and equation (2) we get the intersecting point (x,y) of line AB and line CD. Adding equation (1) & (2) we get #2y=16 or y=8 and x= 9-8=1 :. (x,y)=(1,8)#.

The two end points of altitude CD are #(2,9) and (1,8)#

Length of altitude CD is #d=sqrt((2-1)^2+(9-8)^2)=sqrt2# [Ans]