A triangle has corners A, B, and C located at #(8 ,7 )#, #(1 ,5 )#, and #(6 , 3 )#, 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 #(8   ,7   )#, #(1  ,5    )#, and #(6 ,  3   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Sophia Alfred · Accepted Answer

The end-pts. of Altd. from corner #C# are
#C(6,3) and (270/53,327/53)#.

length of Altd. #=24/sqrt53.# Let #D# be the endpt. of altd. thro. vertex #C(6,3).# Then, #D# lies on the side#AB.# Thus, #D# is the pt. of intersection of altd.#CD# with side #AB# Accordingly, to find #D#, we have to find eqns. of #AB# and #CD# and solve them. Side AB Equation: With #A(8,7)# and, #B(1,5)#, slope of #AB# is #(7-5)/(8-1)=2/7# #B(1,5)# iis on #AB#. Hence, eqn. of #AB# #:y-5=2/7(x-1),# i.e., #y=5+2/7(x-1)............(i)# Equation for Altd. CD: #CD# is perp. to #AB#, and, slope of #AB# is #2/7#, so, the slope of #CD# has to be #-1/(2/7)=-7/2,# & with #C(6,3)# on it, we have, eqn. of #CD : y-3=-7/2(x-6),# or, #y=3-7/2(x-6).................(ii)# To get #D#, solving #(i) & (ii) : 5+2/7(x-1)=3-7/2(x-6)rArr70+4x-4=42-49x+294rArr4x+49x=42+294+4-70rArr53x=270rArrx=270/53# Then, by #(i), y=5+2/7(270/53-1)=5+2/7*217/53=5+62/53=327/53# So, the end-pt. of Altd. from corner #C# is #D(270/53,327/53)#. Finally, length of Altd. #CD# = Perp. Dist. from pt. #C(6,3)# to side #AB : y=5+2/7(x-1)# #=|3-5-2/7(6-1)|/sqrt(1+4/49)=|2+10/7|/sqrt(53/49)=24/sqrt53.#

Caroline Aucoin · Answer

undefined To find the altitude going through corner C of the triangle with vertices A(8, 7), B(1, 5), and C(6, 3), follow these steps:

1. Calculate the slope of the line segment AB.
2. Use the perpendicular slope relationship to find the slope of the altitude.
3. Use the slope of the altitude and the coordinates of point C to write the equation of the altitude in point-slope form.
4. Find the intersection point of the altitude with the line containing side AB to determine the endpoints of the altitude.
5. Calculate the length of the altitude using the distance formula.

Let's proceed with these steps:

1. Slope of AB:
   $ m_{AB} = \frac{y_B - y_A}{x_B - x_A} $

2. Perpendicular slope relationship:
   If the slope of a line is $ m $, the slope of any line perpendicular to it is $ -\frac{1}{m} $.

3. Slope of the altitude:
   Let $ m_{\text{alt}} $ be the slope of the altitude. It will be $ -\frac{1}{m_{AB}} $.

4. Equation of the altitude in point-slope form:
   Using point-slope form $ y - y_1 = m(x - x_1) $, with $ (x_1, y_1) = (6, 3) $ (coordinates of point C) and $ m = -\frac{1}{m_{AB}} $.

5. Intersection point of the altitude with AB:
   Solve the equation of the altitude with the equation of line AB to find the intersection point.

6. Calculate the length of the altitude using the distance formula:
   $ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $, where $ (x_1, y_1) $ and $ (x_2, y_2) $ are the endpoints of the altitude.

By following these steps, you can find the endpoints and length of the altitude going through corner C of the triangle.