A triangle has corners A, B, and C located at #(5 ,2 )#, #(2 ,5 )#, and #(3 ,7 )#, 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 #(5  ,2  )#, #(2  ,5    )#, and #(3 ,7   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Autumn Armstrong · Accepted Answer

Endpoints of altitude #CD# are #(3,7) and (3/2,11/2)#
Length of altitude #CD# is # 2.12# unit

#A(5,2) , B(2,5) , C(3,7)# Let #CD# be the altitude going through #C# touches #D# on line #AB#. #C# and #D# are the endpoints of altitude #CD; CD# is perpendicular on #AB#. Slope of #AB= m_1= (y_2-y_1)/(x_2-x_1)# #=(5-2)/(2-5) = 3/(-3)= -1 :. # Slope of #CD=m_2= -1/m_1= 1 # Equation of line #AB# is # y - y_1 = m_1(x-x_1) #or # y-2 = -1(x-5) or x +y = 7 ; (1) # Equation of line #CD# is # y - y_3 = m_2(x-x_3)# or #y- 7 = 1(x-3) or x -y = -4 (2) # Solving equation (1) and (2) we get the co-ordinates of #D# Adding equation (1) and (2) we get #2 x=3 or x=3/2# #:. y=7-x = 7-3/2= 11/2:. D # is #(3/2,11/2)# Length of altitude #CD# is #CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2) # or #CD = sqrt((3-3/2)^2+(7-11/2)^2)= sqrt 4.5 ~~ 2.12# unit Endpoints of altitude #CD# are #(3,7) and (3/2,11/2)# Length of altitude #CD# is # 2.12# unit [Ans]

James Atkins · Answer

undefined To find the endpoints and length of the altitude going through corner C of the triangle, we first need to find the equation of the line containing side AB of the triangle. Then, we find the perpendicular line passing through point C, which will represent the altitude. The intersection of these two lines will give us the endpoint of the altitude on side AB. Finally, we calculate the length of the altitude using the distance formula.

1. Find the equation of the line containing side AB:
$$ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $$

2. Find the perpendicular line passing through point C:
$$ \text{slope of perpendicular line} = -\frac{1}{\text{slope of side AB}} $$
$$ y - y_C = \text{slope of perpendicular line} \times (x - x_C) $$

3. Solve the system of equations formed by the equations of side AB and the perpendicular line to find the intersection point.

4. Use the distance formula to calculate the length of the altitude.

Let's proceed with the calculations:

$$ \text{Slope of side AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{2 - 5} = \frac{3}{-3} = -1 $$

$$ \text{Equation of side AB:} $$
$$ y - 2 = -1(x - 5) $$
$$ y = -x + 7 $$

$$ \text{Slope of perpendicular line through C} = -\frac{1}{-1} = 1 $$

$$ \text{Equation of perpendicular line through C:} $$
$$ y - 7 = 1(x - 3) $$
$$ y = x + 4 $$

$$ \text{Solving the system of equations:} $$
$$ -x + 7 = x + 4 $$
$$ 2x = 3 $$
$$ x = \frac{3}{2} $$

$$ y = \frac{3}{2} + 4 = \frac{11}{2} $$

$$ \text{Endpoint of altitude on side AB:} (\frac{3}{2}, \frac{11}{2}) $$

$$ \text{Length of altitude} = \sqrt{(x_C - x_{\text{endpoint}})^2 + (y_C - y_{\text{endpoint}})^2} $$
$$ = \sqrt{(3 - \frac{3}{2})^2 + (7 - \frac{11}{2})^2} $$
$$ = \sqrt{(\frac{3}{2})^2 + (\frac{3}{2})^2} $$
$$ = \sqrt{\frac{9}{4} + \frac{9}{4}} $$
$$ = \sqrt{\frac{18}{4}} $$
$$ = \frac{3\sqrt{2}}{2} $$

The endpoints of the altitude on side AB are $(\frac{3}{2}, \frac{11}{2})$ and its length is $\frac{3\sqrt{2}}{2}$.