A triangle has corners A, B, and C located at #(2 ,9 )#, #(1 ,4 )#, and #(6 , 5 )#, 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   ,9   )#, #(1  ,4    )#, and #(6 ,  5   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Mason Adams · Accepted Answer

The endpoints are #(18/13,77/13)# and #(6,5)# and length of the altitude going through corner C is #4.707#

The slope of line #AB# is #(4-9)/(1-2)=(-5)/(-1)=5# and equation of #AB# is #(y-4)=5(x-1)#

i.e. #5x-y=1# ................................(1)

Hence if #CD_|_AB#, with #D# on #AB#, slope of #CD# will be #(-1)/5=-1/5#

and equation of #CD# is #(y-5)=-1/5(x-6)#

or #5y-25=-x+6# i.e. #x+5y=31# ................................(2)

Solving (1) and (2) gives us the coordinates of #D#. For tis putting #y=5x-1# from (1) in (2) gives

#x+25x-5=31# or #26x=36# i.e. #x=36/26=18/13#

and hence #y=5xx18/13-1=77/13#

and coordinated of #D# are #(18/13,77/13)#

and #CD=sqrt((6-18/13)^2+(5-77/13)^2)#

= #sqrt((60/13)^2+(-12/13)^2)#

= #1/13sqrt(3600+144)=sqrt3744/13=61.188/13=4.707#

Violet Aldrich · Answer

undefined To find the endpoints and length of the altitude going through corner C of the triangle, follow these steps:

1. Calculate the slope of the line passing through points A and B using the formula: $ m_{AB} = \frac{{y_B - y_A}}{{x_B - x_A}} $.
2. The altitude passing through corner C is perpendicular to the line AB. Therefore, the slope of the altitude is the negative reciprocal of the slope of line AB, given by $ m_{altitude} = -\frac{1}{{m_{AB}}} $.
3. Use the point-slope form of a line ($ y - y_1 = m(x - x_1) $) with point C to find the equation of the altitude line.
4. Find the intersection point of the altitude line with the line passing through points A and B to determine the endpoints of the altitude.
5. Calculate the distance between point C and the intersection point to find the length of the altitude.