A triangle has corners A, B, and C located at #(3 ,5 )#, #(2 ,9 )#, and #(4 , 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   ,5   )#, #(2  ,9    )#, and #(4 ,  8   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Autumn Armstrong · Accepted Answer

Endpoints #(4,8)# and #( 40/17, 129/17) # and length #7/sqrt{17} #.

Let's continue. Apparently, I am an expert at responding to questions from two-year-olds. The perpendicular to AB through C is the altitude through C. There are a few ways to do this one. We can calculate the slope of AB as #-4,# then the slope of the perpendicular is #1/4# and we can find the meet of the perpendicular through C and the line through A and B. Let's try another way. Let's call the foot of the perpendicular #F(x,y)#. We know the dot product of the direction vector CF with the direction vector AB is zero if they're perpendicular: #(B-A) cdot (F - C) = 0# #(1-,4) cdot (x-4,y-8) = 0# # x - 4 - 4y + 32 = 0 # # x - 4y = -28 # That's one equation. The other equation says #F(x,y)# is on the line through A and B: #(y - 5)(2-3)=(x-3)(9-5)# #5 - y = 4(x-3)# #y = 17 - 4x# When do they meet? #x - 4(17 - 4x) = -28# # x - 68 + 16 x = -28 # # 17 x = 40# # x = 40/17 # # y = 17 - 4 (40/17) = 129/17 # The altitude's length CF is #h = \sqrt{ (40/17-4)^2 + (129/17 - 8)^2} = 7 /sqrt{17}# To verify this, let's solve for the altitude first using the shoelace formula to determine the area: A(3,5), B(2,9), C(4,8) #a = \frac 1 2 | 3(9)-2(5) + 2(8)-9(4) + 4(5)-3(8)| = 7/2# # AB=sqrt{ (3-2)^2+(9-5)^2 } = sqrt{17}# #a = \frac 1 2 b h # # 7/2 = 1/2 h sqrt{17} # # h = 7/sqrt{17} quad quad quad sqrt #

Evelyn Arboleda · Answer

undefined To find the altitude going through corner C of the triangle, you first need to determine the equation of the line containing side AB. Then, find the perpendicular line passing through point C, as the altitude is perpendicular to the side of the triangle it intersects. The intersection point of this perpendicular line and side AB will give you the endpoint of the altitude.

1. Calculate the slope of side AB using points A(3, 5) and B(2, 9).
2. Use the point-slope form of a linear equation to find the equation of the line containing side AB.
3. Determine the negative reciprocal of the slope of side AB to find the slope of the line perpendicular to side AB.
4. Use the slope-intercept form of a linear equation with point C(4, 8) to find the equation of the line passing through point C and perpendicular to side AB.
5. Solve the system of equations formed by the equation of side AB and the perpendicular line to find the intersection point, which will be the endpoint of the altitude.
6. Calculate the length of the altitude using the distance formula between point C and the intersection point found in step 5.