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)

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) (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.
.
The distance formula says:
.

.
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 and . We also recognize that that and .
Substitute these values into the distance formula to get:
.

.
and this simplifies to:
.

.
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
and . It also means that and . Substitute these values
into the distance formula:
.

.
Simplifying terms in the parentheses leads to:
.

.
But we know that d has to equal sqrt(26). Look carefully at it. You should be able to
see that has to be + 25 so that it can add to 1 under the radical and make
the radical be sqrt(26)
.
For to be 25, must either be 5 or -5 so that it squares out to +25.
.
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.
.
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
.
In summary, the answer is that A either equals 9 or -1.
.
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.
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