A triangle has corners A, B, and C located at #(1 ,4 )#, #(2 ,1 )#, and #(5 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
Endpoints of altitude are
length of altitude is
To calculate d the distance between two points, use the distance formula :
Given
As
Hence,
length of the altitude
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To find the endpoint and length of the altitude going through corner C, first, calculate the slope of the line segment AB, then find the equation of the perpendicular line passing through C. The intersection point of this perpendicular line with the line segment AB will be the endpoint of the altitude. Finally, calculate the length of this altitude.

Calculate the slope of AB: [ m_{AB} = \frac{y_B  y_A}{x_B  x_A} ]

The slope of the perpendicular line will be the negative reciprocal of the slope of AB: [ m_{\perp} = \frac{1}{m_{AB}} ]

Using pointslope form, find the equation of the perpendicular line passing through C: [ y  y_C = m_{\perp}(x  x_C) ]

Find the intersection point of this line with the line segment AB, which is the endpoint of the altitude.

Calculate the distance between this endpoint and point C to find the length of the altitude.
Let's proceed with the calculations:

Slope of AB: [ m_{AB} = \frac{1  4}{2  1} = 3 ]

Slope of the perpendicular line: [ m_{\perp} = \frac{1}{3} = \frac{1}{3} ]

Equation of the perpendicular line passing through C: [ y  2 = \frac{1}{3}(x  5) ]
Expanding: [ y  2 = \frac{1}{3}x  \frac{5}{3} ]
[ y = \frac{1}{3}x  \frac{5}{3} + 2 ]
[ y = \frac{1}{3}x + \frac{1}{3} ]
 Finding intersection point with AB:
[ \frac{1}{3}x + \frac{1}{3} = 3x + 7 ]
[ \frac{1}{3}x + 3x = 7  \frac{1}{3} ]
[ \frac{10}{3}x = \frac{20}{3} ]
[ x = 2 ]
[ y = 3(2) + 7 = 1 ]
The intersection point is (2, 1).
 Calculate the length of the altitude:
[ \text{Altitude length} = \sqrt{(5  2)^2 + (2  1)^2} ]
[ \text{Altitude length} = \sqrt{3^2 + 1^2} ]
[ \text{Altitude length} = \sqrt{9 + 1} ]
[ \text{Altitude length} = \sqrt{10} ]
So, the endpoint of the altitude through corner C is (2, 1), and its length is ( \sqrt{10} ).
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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.
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