A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, 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 #(2  ,5   )#, #(7  ,4    )#, and #(6 ,1   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Evelyn Arboleda · Accepted Answer

End points of altitude are #(6,1) and (6.6,4.1)#
Length of altitude is #3.16 (2dp)# unit

Slope of the line AB is #m_1=(y_2-y_1)/(x_2-x_1) = (4-5)/(7-2) = -1/5# Let CD be the altitude from C perpendicular on AB meets at D. Slope of altitude CD is #m_2=-1/(-1/5)=5#[condition of perpendicularity is #m_1*m_2=-1#] Equation of line AB is # y-5 = -1/5(x-2) or 5y +x =27 (1)# Equation of line CD is # y-1 = 5(x-6) or y -5x = -29 (2 )# #5y +x =27 #(1) #5y-25x= -145# (2)*5 Subtracting (2) from (1) we get #26x=172 or x= 86/13 = 6.62# ; #:. y= 5*6.62-29=4.1 :.#End points of altitude are #(6,1) and (6.6,4.1)# Length of altitude is #sqrt((6.6-6)^2+(4.1-1)^2) =3.16#unit[Ans]

Savannah Aguilar · Answer

undefined To find the altitude going through corner C of the triangle, we first need to determine the equation of the line containing side AB. Then, we find the perpendicular line passing through point C. The intersection of these two lines will give us the endpoint of the altitude.

1. Find the slope of side AB using points A and B:
   Slope $ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{7 - 2} = \frac{-1}{5} $.

2. The equation of the line containing side AB can be written as $ y - y_1 = m(x - x_1) $. Choose one of the points, let's say A(2, 5):
   $ y - 5 = \frac{-1}{5}(x - 2) $.

3. Rearrange the equation to slope-intercept form:
   $ y = \frac{-1}{5}x + \frac{11}{5} $.

4. Since the altitude through C is perpendicular to side AB, its slope will be the negative reciprocal of the slope of side AB. Therefore, the slope of the altitude is $ m_{\text{altitude}} = 5 $.

5. Use point C(6, 1) and the slope $ m_{\text{altitude}} = 5 $ to find the equation of the perpendicular line:
   $ y - 1 = 5(x - 6) $.

6. Rearrange the equation to slope-intercept form:
   $ y = 5x - 29 $.

7. To find the intersection point of the two lines (the endpoint of the altitude), solve the system of equations:
   $$
   \begin{cases}
   y = \frac{-1}{5}x + \frac{11}{5} \
   y = 5x - 29
   \end{cases}
   $$

8. Solve the system to find the coordinates of the intersection point, which will be the endpoint of the altitude.

9. Once you have the endpoint, you can calculate the length of the altitude using the distance formula between point C and the endpoint of the altitude.