# A triangle has corners A, B, and C located at #(4 ,3 )#, #(9 ,5 )#, and #(6 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

The end-points are

The triangle's corners are

The altitude's duration is

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To find the endpoints of the altitude going through corner C, we first need to determine the equation of the line containing side AB, which is the line perpendicular to side BC passing through point C.

The slope of side BC is (5-2)/(9-6) = 3/3 = 1. Thus, the slope of the perpendicular line (the altitude) is the negative reciprocal of 1, which is -1.

Using point-slope form with point C (6, 2) and the slope -1, we find the equation of the altitude to be y - 2 = -1(x - 6).

Expanding, we get y - 2 = -x + 6. Rearranging, we have y = -x + 8.

Now, we need to find where this line intersects with side AB.

We set the y-coordinate of A equal to the y-coordinate of the altitude, and solve for x:

3 = -x + 8.

Solving for x, we get x = 5.

Thus, the point where the altitude intersects side AB is (5, 3).

Now, we have the endpoints of the altitude: C(6, 2) and the point of intersection on side AB (5, 3).

To find the length of the altitude, we use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates of the two points:

Distance = √((5 - 6)^2 + (3 - 2)^2) = √((-1)^2 + (1)^2) = √(1 + 1) = √2.

So, the length of the altitude going through corner C is √2.

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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.

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