# A triangle has corners A, B, and C located at #(5 ,2 )#, #(2 ,5 )#, and #(8 ,4 )#, respectively. What are the endpoints and length of the altitude going through corner C?

End points of Altitude CD are

Length of Altitude

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To find the endpoints and length of the altitude going through corner C of the triangle, we first need to determine the equation of the line containing the side AB of the triangle. Then, we can find the point where this line intersects the line passing through point C and perpendicular to side AB, which will give us one endpoint of the altitude.

Let's start by finding the equation of the line passing through points A and B.

The slope of the line AB is given by:

[m_{AB} = \frac{y_B - y_A}{x_B - x_A}]

Substitute the coordinates of points A and B:

[m_{AB} = \frac{5 - 2}{2 - 5} = \frac{3}{-3} = -1]

Using point-slope form, the equation of the line passing through A and B is:

[y - y_A = m_{AB}(x - x_A)]

Substitute the coordinates of point A:

[y - 2 = -1(x - 5)] [y - 2 = -x + 5] [y = -x + 7]

Now, we need to find the equation of the line perpendicular to AB passing through point C. The slope of this line will be the negative reciprocal of the slope of AB, which is (m_{AB} = -1).

Using point-slope form with point C, we have:

[y - y_C = -\frac{1}{m_{AB}}(x - x_C)]

Substitute the coordinates of point C:

[y - 4 = -\frac{1}{-1}(x - 8)] [y - 4 = x - 8] [y = x - 4]

Now, we solve the system of equations formed by the line AB and the line passing through C:

[y = -x + 7] [y = x - 4]

By solving this system, we find the point of intersection which represents one endpoint of the altitude:

[x - 4 = -x + 7] [2x = 11] [x = \frac{11}{2}]

Substitute (x = \frac{11}{2}) into either equation to find (y):

[y = \frac{11}{2} - 4] [y = \frac{11}{2} - \frac{8}{2}] [y = \frac{3}{2}]

So, one endpoint of the altitude is (\left(\frac{11}{2}, \frac{3}{2}\right)).

To find the other endpoint, we use point C, which is ((8, 4)). Therefore, the other endpoint of the altitude is ((8, 4)).

Finally, we calculate the length of the altitude using the distance formula between the two endpoints:

[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]

Substitute the coordinates of the two endpoints:

[d = \sqrt{\left(\frac{11}{2} - 8\right)^2 + \left(\frac{3}{2} - 4\right)^2}] [d = \sqrt{\left(\frac{11}{2} - \frac{16}{2}\right)^2 + \left(\frac{3}{2} - \frac{8}{2}\right)^2}] [d = \sqrt{\left(-\frac{5}{2}\right)^2 + \left(-\frac{5}{2}\right)^2}] [d = \sqrt{\frac{25}{4} + \frac{25}{4}}] [d = \sqrt{\frac{50}{4}}] [d = \sqrt{\frac{25}{2}}] [d = \frac{\sqrt{25}}{\sqrt{2}}] [d = \frac{5}{\sqrt{2}}]

Therefore, the length of the altitude going through corner C is (\frac{5}{\sqrt{2}}) units.

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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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