SOLUTION: If triangle XYZ is isosceles and LINE XY = YZ, find the value(s) of "A" if X(3,4), Y(4,-1)AND Z(A,-2)

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Question 70800: If triangle XYZ is isosceles and LINE XY = YZ, find the value(s) of "A" if X(3,4), Y(4,-1)AND Z(A,-2)
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
To solve this problem use the fact that the distance XY must equal the distance YZ.
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The distance formula says:
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d+=+sqrt%28%28y2+-+y1%29%5E2+%2B+%28x2+-+x1%29%5E2%29
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So to find the distance XY we can use the coordinates at point X ... (3,4) for x1 and y1
and the coordinates at point Y ... (4, -1) for x2 and y2. If we do, then we recognize
that x1+=+3 and y1+=+4. We also recognize that that x2+=+4 and y2+=-1.
Substitute these values into the distance formula to get:
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+d+=+sqrt%28%28-1-4%29%5E2+%2B+%284-3%29%5E2%29
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and this simplifies to:
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d+=+sqrt%28%28-5%29%5E2+%2B+%281%29%5E2%29+=+sqrt%2825+%2B+1%29+=+sqrt%2826%29
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So now we know that side YZ must also have a length of sqrt(26). We will use the distance
formula again. This time let's again use the coordinates at point Y ... (4,-1) for x2 and y2.
And we will use the coordinates at Z ... (A, -2) for x1 and y1. This means that x2+=+4
and y2+=+-1. It also means that x1+=+A and y1+=+-2. Substitute these values
into the distance formula:
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d+=+sqrt%28%28-1+-+%28-2%29%29%5E2+%2B+%284+-+A%29%5E2%29
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Simplifying terms in the parentheses leads to:
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d+=+sqrt%28%281%29%5E2+%2B+%284-A%29%5E2%29+=+sqrt%281+%2B+%284-A%29%5E2%29
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But we know that d has to equal sqrt(26). Look carefully at it. You should be able to
see that %284-A%29%5E2 has to be + 25 so that it can add to 1 under the radical and make
the radical be sqrt(26)
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For %284-A%29%5E2 to be 25, %284-A%29 must either be 5 or -5 so that it squares out to +25.
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For the first case we have 4-A = +5. Subtracting 4 from both sides tells us that -A = +1.
Then to solve for +A multiply both sides of this equation by -1 to get A = -1.
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For the second case we have 4-A = -5. Subtract 4 from both sides to get -A = -9.
Solve for +A by multiplying both sides of the equation by -1 to get A = 9
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In summary, the answer is that A either equals 9 or -1.
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Hope this helps you to see your way through the problem. It might help to sketch the
points on a graph so you can see why either answer for A produces an isosceles triangle
that satisfies the problem.