A triangle has corners A, B, and C located at #(3 ,1 )#, #(6 ,4 )#, and #(9 ,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 #(3  ,1   )#, #(6  ,4    )#, and #(9 ,8   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Sadie Adams · Accepted Answer

If you can't follow the steps step by step, continue reading. Step 1 Find the equation for #AB# #y-y_1=m(x-x_1)#  (point slope formula) #m = (4-1)/(6-3) = 1# (finding the slope using the points #A# and #B#) #y-1=x-3# (plugging #m# and A back into the equation, you could choose A or B and it would be the same)   Segment #AB# lies on the line #y=x-2# Step 2 Find line #l# that is perpendicular to #AB# and intersects with #C#. #y = mx + b# (start with an empty line) #y = -1x + b# (negative reciprocal slope of #AB# for perpendicular line) #8 = -9 + b# (substitute in point #C# to find a line that intersects with #C#) #b = 17# (solve) Now that we have #b# we can list the equation. #l#, our altitude, has the equation #y = -x+17# Step 3 Find point #D# where our altitude #l# and base #AB# intersect #y = -x+17# (#l#)
#y=x-2# (#AB#) So #-x+17=x-2#
#19=2x#
#x = 9.5#
#y = 7.5# (by plugging #x# back into one of the equations) So #D = (9.5,7.5) Step 4 Find the distance of #CD# #sqrt((9-9.5)^2+(8-7.5)^2)# #= sqrt(2)/2# It appears that the answer is a little off. Whoops.

Parker Allen · Answer

undefined The altitude going through corner C of the triangle with corners A, B, and C can be determined by finding the perpendicular line from point C to the line containing segment AB. To find the equation of the line containing segment AB, you first find the slope of AB using the coordinates of points A and B. Then, using the slope and one of the points (A or B), you can find the equation of the line.

Next, you find the negative reciprocal of the slope of AB to get the slope of the altitude. Using point C and the slope of the altitude, you can find the equation of the altitude.

After finding the equation of the altitude, you can solve the system of equations formed by the equation of the altitude and the equation of segment AB to find the point of intersection, which represents the endpoint of the altitude.

Once you have the endpoint of the altitude, you can find its length by calculating the distance between point C and the endpoint using the distance formula.