A triangle has corners A, B, and C located at #(3 ,5 )#, #(2 ,1 )#, and #(5 , 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  ,1    )#, and #(5 ,  8   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Savannah Aguilar · Accepted Answer

#(105/17, 131/17), (65/17, 141/17)#

The vertices of #\Delta ABC# are #A(3, 5)#, #B(2, 1)# & #C(5, 8)#

The area #\Delta# of #\Delta ABC# is given by following formula

#\Delta=1/2|3(1-8)+2(8-5)+5(5-1)|=2.5#

Now, the length of side #AB# is given as

#AB=\sqrt{(3-2)^2+(5-1)^2}=\sqrt17#

If #CN# is the altitude drawn from vertex C to the side AB then the area of #\Delta ABC# is given as

#\Delta =1/2(CN)(AB)#

#2.5=1/2(CN)(\sqrt17)#

#CN=5/\sqrt17#

Let #N(a, b)# be the foot of altitude CN drawn from vertex #C(5, 8)# to the side AB then side #AB# & altitude #CN# will be normal to each other i.e. the product of slopes of AB & CN must be #-1# as follows

#\frac{b-8}{a-5}\times \frac{5-1}{3-2}=-1#

#b=\frac{37-a}{4}\ ............(1)#

Now, the distance formula provides the length of altitude CN.

#\sqrt{(a-5)^2+(b-8)^2}=5/\sqrt17#

#(a-5)^2+(\frac{37-a}{4}-8)^2=(5/\sqrt17)^2#

#a=105/17, 65/17#

Setting above values of #a# in (1), the corresponding values of #b# respectively are

#b=131/17, 141/17#

hence, the end points of altitude #CN# are

#(105/17, 131/17), (65/17, 141/17)#

Noah Atkinson · 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 segment AB using the formula:
$$ m_{AB} = \frac{y_B - y_A}{x_B - x_A} $$

2. Use the point-slope form of a line to find the equation of the line perpendicular to AB passing through point C:
$$ y - y_C = -\frac{1}{m_{AB}}(x - x_C) $$

3. Substitute the coordinates of point C and the slope of AB into the equation to find the equation of the altitude line.

4. Find the intersection point of the altitude line and the line segment AB by solving the system of equations formed by the altitude line and the line segment AB.

5. The intersection point gives the endpoints of the altitude.

6. Calculate the length of the altitude using the distance formula between the intersection point and point C.

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