# A triangle has corners A, B, and C located at #(2 ,2 )#, #(3 ,4 )#, and #(6 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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To find the endpoints and length of the altitude going through corner C of the triangle with vertices A(2, 2), B(3, 4), and C(6, 8), we first need to determine the slope of the line segment AB and then find the perpendicular slope to this segment, which will be the slope of the altitude.

The slope of AB is given by: [ m_{AB} = \frac{y_B - y_A}{x_B - x_A} ]

[ m_{AB} = \frac{4 - 2}{3 - 2} = \frac{2}{1} = 2 ]

The perpendicular slope to AB, which is the slope of the altitude, is the negative reciprocal of the slope of AB: [ m_{\text{altitude}} = -\frac{1}{m_{AB}} = -\frac{1}{2} ]

Now that we have the slope of the altitude, we can use point-slope form to find the equation of the line passing through C(6, 8) with slope -1/2: [ y - y_C = m_{\text{altitude}}(x - x_C) ] [ y - 8 = -\frac{1}{2}(x - 6) ] [ y - 8 = -\frac{1}{2}x + 3 ] [ y = -\frac{1}{2}x + 11 ]

Next, we find the intersection point of this line with the line segment AB. Substitute the x-value of this point into the equation of the altitude to find the corresponding y-value: [ -\frac{1}{2}x + 11 = 2x - 2 ] [ \frac{5}{2}x = 13 ] [ x = \frac{26}{5} ]

Substitute ( x = \frac{26}{5} ) into the equation of the altitude to find y: [ y = -\frac{1}{2}(\frac{26}{5}) + 11 = \frac{49}{5} ]

Therefore, the endpoints of the altitude going through corner C are ( (\frac{26}{5}, \frac{49}{5}) ) and ( (6, 8) ). To find the length of the altitude, use the distance formula between these endpoints: [ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] [ \text{Length} = \sqrt{(\frac{26}{5} - 6)^2 + (\frac{49}{5} - 8)^2} ] [ \text{Length} = \sqrt{(\frac{16}{5})^2 + (\frac{9}{5})^2} ] [ \text{Length} = \sqrt{\frac{256}{25} + \frac{81}{25}} ] [ \text{Length} = \sqrt{\frac{337}{25}} ] [ \text{Length} = \frac{\sqrt{337}}{5} ]

So, the endpoints of the altitude are ( (\frac{26}{5}, \frac{49}{5}) ) and ( (6, 8) ), and the length of the altitude is ( \frac{\sqrt{337}}{5} ).

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