A triangle has corners A, B, and C located at #(7 ,3 )#, #(4 ,8 )#, and #(3 , 4 )#, respectively. What are the endpoints and length of the altitude going through corner C?
The answer is (5.5,5.5)
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To find the endpoints of the altitude going through corner C of the triangle ABC, we first need to find the equation of the line containing side AB. Then, we'll find the midpoint of AB, which will be one endpoint of the altitude. The other endpoint will be corner C itself. Finally, we'll calculate the length of this altitude.

Find the equation of the line containing side AB:
 Using the coordinates of points A and B, calculate the slope of AB using the formula: [ m = \frac{y_2  y_1}{x_2  x_1} ]
 Then, use the slopeintercept form of a line equation, ( y = mx + b ), where ( m ) is the slope and ( b ) is the yintercept. Substitute one of the points, say A, into the equation to find ( b ).

Find the midpoint of AB:
 Use the midpoint formula: [ M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]

The altitude through corner C will be perpendicular to side AB, so its slope will be the negative reciprocal of the slope of AB.

Use the pointslope form of a line equation, ( y  y_1 = m(x  x_1) ), where ( m ) is the slope and ( (x_1, y_1) ) is a point on the line (the midpoint of AB).

Substitute the coordinates of point C into the equation to find the equation of the altitude line.

Finally, calculate the length of the altitude by finding the distance between the endpoints using the distance formula: [ d = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ]
Performing these steps will yield the endpoints and length of the altitude going through corner C.
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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.
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