# A triangle has corners A, B, and C located at #(2 ,3 )#, #(5 ,8 )#, and #(3 , 4 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Now, the distance formula provides the length of altitude CN.

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To find the endpoints and length of the altitude going through corner C of the triangle with vertices A, B, and C, located at (2, 3), (5, 8), and (3, 4) respectively, follow these steps:

- Calculate the slope of the line passing through A and B.
- Find the equation of the line passing through A and B using the point-slope form.
- Determine the perpendicular line passing through point C, which represents the altitude.
- Find the intersection point of this perpendicular line with the line segment AB to get the endpoint of the altitude.
- Calculate the length of the altitude using the distance formula between point C and the endpoint found in step 4.

Let's proceed with the calculations:

Step 1: Calculate the slope of line AB: [ m_{AB} = \frac{y_B - y_A}{x_B - x_A} ]

Step 2: Find the equation of line AB using point-slope form: [ y - y_A = m_{AB}(x - x_A) ]

Step 3: Determine the slope of the line perpendicular to line AB: [ m_{\perp} = -\frac{1}{m_{AB}} ]

Step 4: Find the equation of the perpendicular line passing through point C: [ y - y_C = m_{\perp}(x - x_C) ]

Step 5: Find the intersection point of the perpendicular line with line segment AB: [ x = \frac{x_A + x_B}{2}, \quad y = \frac{y_A + y_B}{2} ]

Step 6: Calculate the length of the altitude using the distance formula: [ d = \sqrt{(x_C - x_{\text{endpoint}})^2 + (y_C - y_{\text{endpoint}})^2} ]

Now, substitute the given coordinates into the formulas and calculate the values accordingly to find the endpoints and length of the altitude going through corner C.

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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.

- A line segment is bisected by line with the equation # 3 y - 3 x = 4 #. If one end of the line segment is at #(2 ,4 )#, where is the other end?
- A triangle has corners A, B, and C located at #(2 ,2 )#, #(3 ,4 )#, and #(6 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- A triangle has corners A, B, and C located at #(2 ,7 )#, #(5 ,3 )#, and #(9 , 4 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- What is the centroid of a triangle with corners at #(4,1 )#, #(6,3 )#, and #(5 , 1 )#?
- A triangle has corners A, B, and C located at #(1 ,8 )#, #(6 ,3 )#, and #(7 ,4 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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