Two corners of an isosceles triangle are at #(9 ,4 )# and #(3 ,8 )#. If the triangle's area is #48 #, what are the lengths of the triangle's sides?
Three sides of the triangle are
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The lengths of the sides of the isosceles triangle are 10 and 10.
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To find the lengths of the sides of the triangle, we can use the distance formula to find the distances between the given points. Let's denote the coordinates of the third vertex of the triangle as (x, y).

Find the base length (the distance between the two given points): Distance Formula: (d = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}) Substituting the given points: (d = \sqrt{(3  9)^2 + (8  4)^2} = \sqrt{36 + 16} = \sqrt{52})

The area of the triangle can be calculated using the formula: (Area = \frac{1}{2} \times \text{base} \times \text{height}) Substituting the known values: (48 = \frac{1}{2} \times \sqrt{52} \times \text{height}) Solving for height: (96 = \sqrt{52} \times \text{height}) (\text{height} = \frac{96}{\sqrt{52}})

Now, using the Pythagorean theorem, we can find the distance from the third vertex to the midpoint of the base (which is the height of the triangle): (48^2 = (\frac{\sqrt{52}}{2})^2 + (\text{height})^2) Solving for height: (48^2  (\frac{\sqrt{52}}{2})^2 = (\text{height})^2) (\text{height} = \sqrt{48^2  (\frac{\sqrt{52}}{2})^2})

Finally, once we have the height, we can use the distance formula again to find the distance from the third vertex to one of the given vertices (since it's an isosceles triangle): (d = \sqrt{(9  x)^2 + (4  y)^2}) or (d = \sqrt{(3  x)^2 + (8  y)^2})
Using the given area, we solve for height, and then we can find the lengths of the sides using the distances we calculated.
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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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