A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, and #(6 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?
End points of altitude are
Length of altitude is
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To find the altitude going through corner C of the triangle, we first need to determine the equation of the line containing side AB. Then, we find the perpendicular line passing through point C. The intersection of these two lines will give us the endpoint of the altitude.

Find the slope of side AB using points A and B: Slope ( m = \frac{y_2  y_1}{x_2  x_1} = \frac{4  5}{7  2} = \frac{1}{5} ).

The equation of the line containing side AB can be written as ( y  y_1 = m(x  x_1) ). Choose one of the points, let's say A(2, 5): ( y  5 = \frac{1}{5}(x  2) ).

Rearrange the equation to slopeintercept form: ( y = \frac{1}{5}x + \frac{11}{5} ).

Since the altitude through C is perpendicular to side AB, its slope will be the negative reciprocal of the slope of side AB. Therefore, the slope of the altitude is ( m_{\text{altitude}} = 5 ).

Use point C(6, 1) and the slope ( m_{\text{altitude}} = 5 ) to find the equation of the perpendicular line: ( y  1 = 5(x  6) ).

Rearrange the equation to slopeintercept form: ( y = 5x  29 ).

To find the intersection point of the two lines (the endpoint of the altitude), solve the system of equations: [ \begin{cases} y = \frac{1}{5}x + \frac{11}{5} \ y = 5x  29 \end{cases} ]

Solve the system to find the coordinates of the intersection point, which will be the endpoint of the altitude.

Once you have the endpoint, you can calculate the length of the altitude using the distance formula between point C and the endpoint of the altitude.
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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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