# An isosceles triangle has sides #a,# #b,# and #c# with sides #a# and #c# being equal in length. If side #b# goes from #(5 ,1 )# to #(3 ,2 )# and the triangle's area is #8 #, what are the possible coordinates of the triangle's third vertex?

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To find the possible coordinates of the triangle's third vertex, we need to use the formula for the area of a triangle:

Area = (1/2) * base * height

Given that side b goes from (5,1) to (3,2), we can calculate the length of side b using the distance formula:

Length of side b = sqrt((5 - 3)^2 + (1 - 2)^2) = sqrt(4 + 1) = sqrt(5)

Now, we know the length of side b and the area of the triangle. We can rearrange the formula for the area of a triangle to solve for the height:

Height = (2 * Area) / base

Since the triangle is isosceles, the base (side b) is equal to the length of side b, which is sqrt(5). Substituting the given values:

Height = (2 * 8) / sqrt(5) = 16 / sqrt(5)

Now, we can find the possible coordinates of the third vertex by moving along the line that is perpendicular to side b and passes through its midpoint.

First, find the midpoint of side b:

Midpoint = ((5 + 3) / 2, (1 + 2) / 2) = (4, 1.5)

Next, determine the slope of side b:

Slope of side b = (2 - 1) / (3 - 5) = 1 / (-2) = -1/2

The negative reciprocal of the slope of side b will give us the slope of the line perpendicular to side b:

Perpendicular slope = -1 / (-1/2) = 2

Using the slope and the midpoint, we can find the equation of the line perpendicular to side b:

y - 1.5 = 2(x - 4)

Now, we can find the possible coordinates of the third vertex by substituting values for x into this equation and solving for y.

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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.

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