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?
Endpoints of altitude
Length of altitude
By signing up, you agree to our Terms of Service and Privacy Policy
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.

Find the equation of the line containing side AB: [ y  y_1 = \frac{y_2  y_1}{x_2  x_1}(x  x_1) ]

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) ]

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

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}).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 A line segment is bisected by a line with the equation #  3 y + 4 x = 6 #. If one end of the line segment is at #( 3 , 1 )#, where is the other end?
 A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,9 )#, and #(1 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
 A triangle has corners at #(8 , 4 )#, ( 4 , 3)#, and #( 6 , 5 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?
 Construct a regular pentagon using a compass and straight edge? Explain each step
 What is a perpendicular bisector?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7