What is the orthocenter of a triangle with corners at #(7 ,3 )#, #(4 ,8 )#, and (6 ,8 )#?
The orthocenter is
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To find the orthocenter of a triangle, we need to determine the point where the three altitudes of the triangle intersect. The altitude of a triangle is a perpendicular line segment from one vertex to the opposite side.
Given the vertices of the triangle as (7, 3), (4, 8), and (6, 8), we first find the slopes of the lines passing through each pair of vertices to determine the equations of the altitudes.
The slopes of the lines passing through the points are:

Slope of the line passing through (7, 3) and (4, 8): m1 = (8  3) / (4  7) = 5 / 3 = 5/3

Slope of the line passing through (7, 3) and (6, 8): m2 = (8  3) / (6  7) = 5 / 1 = 5

Slope of the line passing through (4, 8) and (6, 8): m3 = (8  8) / (6  4) = 0
Now, using the pointslope form of a linear equation, we can find the equations of the altitudes:

Equation of the altitude passing through (7, 3): y  3 = (5/3)(x  7)

Equation of the altitude passing through (4, 8): y  8 = (5)(x  4)

Equation of the altitude passing through (6, 8): y  8 = 0 (since the line is horizontal)
Next, we find the point of intersection of these lines, which represents the orthocenter of the triangle. Solving the system of equations formed by the three altitude lines will give us the coordinates of the orthocenter.
After solving the system, we find the orthocenter at the point of intersection, which gives the coordinates of the orthocenter of the triangle.
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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 triangle has corners A, B, and C located at #(5 ,2 )#, #(2 ,5 )#, and #(8 ,7 )#, respectively. What are the endpoints and length of the altitude going through corner C?
 What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(2 ,4 )#, and (8 ,6 )#?
 What is the orthocenter of a triangle with corners at #(3 ,3 )#, #(2 ,4 )#, and (7 ,9 )#?
 A line segment is bisected by a line with the equation # 3 y  8 x = 2 #. If one end of the line segment is at #(1 ,3 )#, where is the other end?
 A triangle has corners A, B, and C located at #(5 ,5 )#, #(3 ,9 )#, and #(4 , 1 )#, respectively. What are the endpoints and length of the altitude going through corner C?
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