A triangle has corners A, B, and C located at #(4 ,7 )#, #(3 ,5 )#, and #(6 ,1 )#, 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 #(4  ,7  )#, #(3  ,5    )#, and #(6 ,1   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Aaliyah Anderson · Accepted Answer

(2,3) ,length of altitude #2sqrt5#

The altitude from the corner C would be perpendicular to line segment AB. Slope of AB would be #(7-5)/(4-3) = 2#. Therefore slope of the perpendicular would be #- 1/2#. Since this perpendicular line passes through corner C (6,1), its equation in point slope form would be #y-1= -1/2 (x-6)#

In a similar vein, line AB's equation would be y-7=2(x-4).

The intersection of line AB and the perpendicular line, whose equations have been worked out as above, would be the altitude's end point.

From the second equation, y=7+2x-8 = 2x-1. Substitute this for y in the first equation to get #2x-1-1= -1/2 x +3# Or, #2x +1/2 x= 3+2# #5x/2 =5 -> x=2# , then y= 2(2)-1=3

Therefore, the necessary endpoint would be (2,3)

Length of the altitude would be #sqrt((6-2)^2 +(1-3)^2)= sqrt20=2sqrt5#

Sophia Alfred · Answer

undefined To find the endpoints of the altitude going through corner C, we first need to determine the slope of the line segment AB, which is the side opposite to corner C. Then, we find the negative reciprocal of this slope to get the slope of the altitude. After that, we use the point-slope form of a line to find the equation of the altitude passing through corner C. Finally, we solve for the intersection point of this altitude with the line segment AB to get the endpoints of the altitude.

The slope of the line segment AB is (5 - 7) / (3 - 4) = -2 / -1 = 2. The negative reciprocal of this slope is -1/2, which is the slope of the altitude passing through corner C.

Using the point-slope form of a line with the point (6, 1) and slope -1/2, we have the equation of the altitude:

y - 1 = -1/2(x - 6).

Now, we need to find the intersection point of this altitude with the line segment AB. We do this by setting the equations of the altitude and the line segment AB equal to each other and solving for x and y.

Upon solving, we find the intersection point to be (4.4, 2.8).

So, the endpoints of the altitude going through corner C are (4.4, 2.8). Now, to find the length of the altitude, we calculate the distance between corner C and the intersection point using the distance formula:

Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
         = √[(4.4 - 6)^2 + (2.8 - 1)^2]
         ≈ √[1.6^2 + 1.8^2]
         ≈ √(2.56 + 3.24)
         ≈ √5.8
         ≈ 2.41 (approximately).

Therefore, the length of the altitude going through corner C is approximately 2.41 units.