A triangle has corners A, B, and C located at #(4 ,7 )#, #(3 ,2 )#, and #(2 ,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  ,2    )#, and #(2 ,1   )#, respectively. What are the endpoints and length of the altitude going through corner C?

James Atkins · Accepted Answer

The coordinates of end point of altitude is #(2,1) and (36/13,11/13)# and length of altitude is #0.784# unit

#A(4,7) , B(3,2) , C(2,1)# 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)# #=(2-7)/(3-4) = 5 :. # Slope of #CD=m_2= -1/m_1= - 1/5# Equation of line #AB# is # y - y_1 = m_1(x-x_1) # or # y- 7 = 5(x-4) or 5x-y = 13 ; (1) # Equation of line #CD# is # y - y_3 = m_2(x-x_3)# or #y- 1 = -1/5(x-2) or x+5y=7 (2)# Mutiplying equation (2) by #5# we get # 5x+25y=35 (3)# Subtracting equation (1) from equation(3) we get , #26y=22 or y =22/26=11/13 # #:. x= 7-5y=7- 55/13=36/13 :. # The co-ordinates of point #D# is#(36/13,11/13)# and the coordinates of end point of altitude is #(2,1) and (36/13,11/13)# Length of altitude #CD# is #CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2) # or #CD = sqrt((2-36/13)^2+(1-11/13)^2)~~0.784# unit [Ans]

Autumn Armstrong · Answer

undefined To find the altitude going through corner C of the triangle with vertices A(4, 7), B(3, 2), and C(2, 1), we first need to determine the equation of the line containing side AB, as it is perpendicular to the altitude we seek. Then, we can find the intersection point of this line with the altitude from point C, which will give us the endpoint of the altitude.

The equation of the line passing through points A(4, 7) and B(3, 2) can be found using the point-slope form: $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is a point on the line. The slope $ m $ can be found using the formula $ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} $ for two points $ (x_1, y_1) $ and $ (x_2, y_2) $.

After finding the equation of the line containing side AB, we can determine its intersection with the altitude from point C. This intersection point will be one endpoint of the altitude. Then, using the distance formula, we can find the length of the altitude from point C to this intersection point.

Here are the steps summarized:

1. Find the slope of side AB using points A and B.
2. Use the point-slope form to find the equation of the line containing side AB.
3. Since the altitude from point C is perpendicular to side AB, its slope will be the negative reciprocal of the slope of side AB.
4. Find the equation of the line passing through point C with the slope determined in step 3.
5. Solve the system of equations formed by the equations of side AB and the altitude from point C to find the intersection point, which will be an endpoint of the altitude.
6. Use the distance formula to find the length of the altitude from point C to the intersection point.

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