A triangle has corners A, B, and C located at #(1 ,6 )#, #(9 ,3 )#, and #(2 ,2 )#, 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 #(1  ,6  )#, #(9  ,3    )#, and #(2 ,2  )#, respectively. What are the endpoints and length of the altitude going through corner C?

Violet Aldrich · Accepted Answer

The altitude endpoints are #(2,2)# and #(3.1918,5.1781)#
The altitude has a length of #3.394#

The line segment #AB# through #A=(1,6)# and #B=(9,3)#
has a slope of #m_(AB)=(6-3)/(1-9)=-3/8#
and an equation of #y-6=-3/8(x-1)#
or
#color(white)("XXX")y=(-3x+51)/8#

In #triangleABC# the altitude through #C# is a line segment perpendicular to #AB# and therefore has a slope of
#color(white)("XXX")m_"alt"=8/3#

Since the altitude passes through #C=(2,2)# and has a slope of #m_"alt"=8/3#
it's line has an equation:
#color(white)("XXX")y-2=8/3(x-2)#
or
#color(white)("XXX")y=(8x-10)/3#

and
#color(white)("XXX")"Altitude": y=(8x-10)/3#
intersect at #(3.1918,5.1781)# (approximate)

The length of the altitude from #C: (2,2)# to the point of intersection is
#color(white)("XXX")d=sqrt((2-3.1918)^2+(2-5.1781)^2)~~5.1781#

Savannah Aguilar · Answer

undefined The endpoints of the altitude going through corner C are (2,2) and the foot of the altitude, which we'll call point D. The length of the altitude can be calculated using the distance formula between points C and D.