An isosceles triangle has sides #a, b# and #c# with sides #b# and #c# being equal in length. If side #a# goes from #(2 ,9 )# to #(8 ,5 )# and the triangle's area is #64 #, what are the possible coordinates of the triangle's third corner?
Let's do this directly. We have
Two choices for the third vertex:
#(x,y) = (5,7) pm (32/13) (4,6) = (63/13, 101/13) or (193/13, 283/13)#
Let's check one:
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To find the possible coordinates of the triangle's third corner, we first need to determine the lengths of sides (b) and (c), which are equal, using the given coordinates. Then we can use the formula for the area of a triangle to solve for the possible coordinates of the third corner.

Calculate the length of side (a) using the distance formula: [ a = \sqrt{(8  2)^2 + (5  9)^2} ]

Since the triangle is isosceles, sides (b) and (c) have the same length as side (a).

Given the area of the triangle ((64)), we can use the formula for the area of a triangle: [ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ] [ 64 = \frac{1}{2} \times a \times h ] [ h = \frac{2 \times 64}{a} ]

Now, we can use the height and the midpoint of side (a) to find the coordinates of the third corner.

Calculate the midpoint of side (a): [ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Find the equation of the line perpendicular to side (a) passing through its midpoint: [ \text{Slope of } a = m_a = \frac{y_2  y_1}{x_2  x_1} ] [ \text{Slope of perpendicular line} = \frac{1}{m_a} ]

Use the slope of the perpendicular line and the midpoint to find the equation of the line: [ y  y_{\text{midpoint}} = m_{\text{perpendicular}}(x  x_{\text{midpoint}}) ]

Substitute the calculated midpoint coordinates and slope into the equation and solve for (y) to find the possible coordinates of the third corner.

Repeat the process for the negative slope of the perpendicular line to find the other possible coordinates.
By following these steps, you can determine the possible coordinates of the triangle's third corner.
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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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