An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,1 )# to #(8 ,5 )# and the triangle's area is #27 #, what are the possible coordinates of the triangle's third corner?
Let's begin by finding the length of side "a":
Use side "a" as the base of the triangle, find the height:
We can used the distance formula to write one equation for the two possible points:
Square both sides:
We need one more equation. Find the slope of side "a":
We can use the slope and the point to find the equation of line for the height:
The equation for the height is:
Solving equations [1] and [2] is very long and tedious process. I gave them to WolframAlpha. Here are the points:
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Given the coordinates of two vertices of the triangle, you can find the length of side (A) using the distance formula. Then, you can use the formula for the area of a triangle to find the height corresponding to side (A). Once you have the height, you can determine the possible coordinates of the third corner by considering the fact that it lies on a line parallel to side (A) and at a distance equal to the height from the line segment connecting the given two vertices.
Here are the steps:

Calculate the length of side (A) using the distance formula: [ A = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ] Substitute the coordinates: ( (7,1) ) and ( (8,5) ).

Use the formula for the area of a triangle: [ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ] Substitute the known values: (\text{Area} = 27) and (\text{base} = A).

Solve for the height: [ 27 = \frac{1}{2} \times A \times \text{height} ] Solve for (\text{height}).

Determine the possible coordinates of the third corner: Since the triangle is isosceles, the third corner lies on a line parallel to side (A) and at a distance equal to the height from the line segment connecting the given two vertices. You can find the equation of the line parallel to side (A) passing through one of the given vertices and then use the distance and direction to find the coordinates of the third corner.

Repeat step 4 for both given vertices to find the possible coordinates of the 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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 Which line is perpendicular to the line 2x + 3y = 12 ?
 Circle A has a center at #(2 ,5 )# and a radius of #3 #. Circle B has a center at #(7 ,2 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
 What is the perimeter of a triangle with corners at #(3 ,4 )#, #(4 ,8 )#, and #(8 ,7 )#?
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