What is the orthocenter of a triangle with corners at #(7 ,3 )#, #(4 ,8 )#, and (6 ,3 )#?
The orthocenter is
Please notice that the slope of line is 0, therefore, the altitude will be a vertical line:
The slope, n, of the altitudes will be the slope of a perpendicular line:
Substitute the x value from equation [1] into equation [2] to find the y coordinate of the orthocenter:
By signing up, you agree to our Terms of Service and Privacy Policy
To find the orthocenter of a triangle, we need to find the point where the three altitudes of the triangle intersect. An altitude is a line segment that extends from one vertex of the triangle perpendicular to the opposite side.
Given the vertices of the triangle as ( (7, 3) ), ( (4, 8) ), and ( (6, 3) ), we can proceed as follows:
 Find the slopes of the lines passing through each pair of vertices.
 Find the equations of the perpendicular bisectors of each side of the triangle.
 Solve the system of equations formed by the perpendicular bisectors to find their point of intersection, which is the orthocenter.
Let's calculate the orthocenter:

Slopes of the sides:
 Slope of ( (7, 3) ) to ( (4, 8) ): ( m_1 = \frac{83}{47} = \frac{5}{3} )
 Slope of ( (7, 3) ) to ( (6, 3) ): ( m_2 = \frac{33}{67} = 0 )
 Slope of ( (4, 8) ) to ( (6, 3) ): ( m_3 = \frac{38}{64} = \frac{5}{2} )

Perpendicular bisectors:
 Midpoint ( M_1 ) of ( (7, 3) ) and ( (4, 8) ): ( (\frac{7+4}{2}, \frac{3+8}{2}) = (5.5, 5.5) ) The perpendicular bisector of ( (7, 3) ) and ( (4, 8) ) is the line perpendicular to the segment ( (7, 3) )  ( (4, 8) ) passing through ( (5.5, 5.5) ). Its equation is ( x = 5.5 ).
 Midpoint ( M_2 ) of ( (7, 3) ) and ( (6, 3) ): ( (\frac{7+6}{2}, \frac{3+3}{2}) = (6.5, 3) ) The perpendicular bisector of ( (7, 3) ) and ( (6, 3) ) is the line perpendicular to the segment ( (7, 3) )  ( (6, 3) ) passing through ( (6.5, 3) ). Its equation is ( y = 3 ).
 Midpoint ( M_3 ) of ( (4, 8) ) and ( (6, 3) ): ( (\frac{4+6}{2}, \frac{8+3}{2}) = (5, 5.5) ) The perpendicular bisector of ( (4, 8) ) and ( (6, 3) ) is the line perpendicular to the segment ( (4, 8) )  ( (6, 3) ) passing through ( (5, 5.5) ). Its equation is ( x = 5 ).

The orthocenter is the intersection of the perpendicular bisectors. Since the equations of two of the bisectors are vertical and horizontal lines, the intersection point must lie on both. Thus, the orthocenter is ( (5.5, 3) ).
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.
 What is the centroid of a triangle with corners at #(2 , 5 )#, #(4 , 9 )#, and #(3 , 3 )#?
 A triangle has corners A, B, and C located at #(4 ,7 )#, #(9 ,5 )#, and #(6 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
 A triangle has corners A, B, and C located at #(5 ,5 )#, #(7 ,9 )#, and #(9 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?
 A triangle has corners A, B, and C located at #(1 ,1 )#, #(3 ,4 )#, and #(2 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
 Draw a line l and a point A not lying on l. Draw the line m that is passing through A and parallel to #l#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7