A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,9 )#, and #(4 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
Length of altitude passing through point C = 3.0309#
Equation of side AB
Let Slope of side AB be ‘m’
Slope of perpendicular line to AB is
Eqn of Altitude to AB passing through point C is
Solving Eqns (1) & (2) we get the base of the altitude passing through point C.
Solving the two equations, we get
Length of the altitude passing through point C
Length of Altitude passing through point C = 3.0509#
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To find the endpoints of the altitude going through corner C of the triangle, you need to first determine the equation of the line containing side AB, then find the perpendicular line passing through point C. The point where these two lines intersect will be one endpoint of the altitude.

Find the slope of side AB using points A and B. Slope of AB = (y2  y1) / (x2  x1) = (9  6) / (3  5) = 3 / 2 = 3/2

Using the pointslope form, find the equation of the line containing side AB. y  y1 = m(x  x1), where m is the slope and (x1, y1) is any point on the line. Using point A(5, 6): y  6 = (3/2)(x  5) y  6 = (3/2)x + 15/2 y = (3/2)x + 15/2 + 6 y = (3/2)x + 15/2 + 12/2 y = (3/2)x + 27/2

Determine the perpendicular slope to side AB, which will be the negative reciprocal of 3/2. Perpendicular slope = 2/3

Using point C(4, 2) and the perpendicular slope, find the equation of the line passing through C. y  y1 = m(x  x1), where m is the perpendicular slope and (x1, y1) is any point on the line. Using point C(4, 2): y  2 = (2/3)(x  4) y  2 = (2/3)x  8/3 y = (2/3)x  8/3 + 6/3 y = (2/3)x  2/3

Now, solve the system of equations formed by the line containing side AB and the line passing through C to find the intersection point, which will be one endpoint of the altitude.
(3/2)x + 27/2 = (2/3)x  2/3 Multiply both sides by 6 to clear the fractions: 9x + 81 = 4x  4
13x = 85 x = 85/13
Substitute x back into either equation to find y: y = (2/3)(85/13)  2/3 y ≈ 130/13

Therefore, one endpoint of the altitude is approximately (85/13, 130/13).
To find the other endpoint, you can use the distance formula between the intersection point and point C. Then double the distance to find the length 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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