# A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, and #(2 ,3 )#, respectively. What are the endpoints and length of the altitude going through corner C?

The altitude basis coordinates over the side

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The altitude going through corner C of the triangle with corners A(2, 5), B(7, 4), and C(2, 3) has endpoints C and the foot of the altitude. The foot of the altitude is the point where the altitude intersects the side opposite to corner C.

To find the foot of the altitude, we need to find the equation of the line passing through A and B, which is the side opposite to corner C. Then we find the perpendicular line passing through C, which will be the altitude. The intersection point of this perpendicular line with the line passing through A and B gives us the foot of the altitude.

First, calculate the slope of the line passing through A and B using the formula: [ m = \frac{{y_B - y_A}}{{x_B - x_A}} ]

Then, find the negative reciprocal of this slope to get the slope of the perpendicular line, which is the slope of the altitude.

Next, use the point-slope form of a line to find the equation of the perpendicular line passing through C.

After finding the equation of the perpendicular line, solve it simultaneously with the line passing through A and B to find the intersection point, which is the foot of the altitude.

Finally, you'll have the endpoints of the altitude and can calculate its length using the distance formula.

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When evaluating a one-sided 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 one-sided 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 one-sided 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 #(2 ,7 )#, #(5 ,3 )#, and #(2 , 4 )#, respectively. What are the endpoints and length of the altitude going through corner C?
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- What is the orthocenter of a triangle with corners at #(3 ,6 )#, #(3 ,2 )#, and (5 ,7 )#?
- A triangle has corners A, B, and C located at #(4 ,3 )#, #(7 ,4 )#, and #(2 ,5 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- Given #x^2 + 4x - 32 = 0# use geometric construction to determine roots? Generalize to #ax^2+bx+c=0# finding a geometric interpretation of a quadratic formula #(-b+- sqrt(b^2-4ac))/(2a)#?

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