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

Claire Akin · Accepted Answer

The end points of the altitude is #=(13/2,15/2)# and the length is #=1.58#

The triangle's corners are

#A=(5,3)#

#B=(7,9)#

#C=(5,8)#

The slope of the line #AB# is #m=(9-3)/(7-5)=3#

The equation of line #AB# is

#y-3=3(x-5)#

#y-3=3x-15#

#y-3x=-12#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-1/3#

The equation of the altitude through #C# is

#y-8=-1/3(x-5)#

#3y-24=-x+5#

#3y+x=5+24=29#................................#(2)#

Solving for #x# and #y# in equations #(1)# and #(2)#, we get

#3*(3x-12)+x=29#

#9x-36+x=29#

#10x=29+36#

#x=65/10=13/2#

#y=(3*13/2-12)=15/2#

The end points of the altitude is #=(13/2,15/2)#

The altitude's duration is

#=sqrt((5-13/2)^2+(8-15/2)^2)#

#=sqrt((3/2)^2+(1/2)^2)#

#=sqrt(10)/2#

#=1.58#

Ella Armstrong · Answer

undefined The endpoints of the altitude going through corner C can be found by determining the perpendicular line passing through point C and intersecting the line containing side AB.

First, find the slope of the line containing side AB using points A and B. Then, determine the negative reciprocal of this slope to find the slope of the perpendicular line passing through point C. Next, use the point-slope form to find the equation of this perpendicular line. The intersection of this perpendicular line with side AB gives the endpoints of the altitude.

Once you have the endpoints, you can calculate the length of the altitude using the distance formula.

1. Find the slope of AB using points A and B.
2. Determine the negative reciprocal of the slope of AB to find the slope of the altitude passing through point C.
3. Use the point-slope form to find the equation of the altitude passing through point C.
4. Find the intersection of this line with the line containing side AB to get the endpoints.
5. Calculate the length of the altitude using the distance formula.