SOLUTION: Find the missing side in the triangle using pythagorean theorem, give answer as a simplified radical x,2x,15 "Find X"

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Question 85290: Find the missing side in the triangle using pythagorean theorem, give answer as a simplified radical
x,2x,15
"Find X"
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given: A triangle having the sides x, 2x, and 15. The problem says to use the Pythagorean
theorem to find the value of x. (This means that the given triangle is a right triangle
since the Pythagorean theorem only applies to right triangles.)
.
The way this problem is stated, there are two possible answers depending on which side you
assume is the long side (hypotenuse). The side x cannot be the hypotenuse because it is shorter
than the side 2x. Therefore, there are two possibilities for the hypotenuse. Either the
hypotenuse is 2x or it is 15.
.
Let's first assume that the hypotenuse is 2x. That means that one leg is x and the other is
15.
.
By the Pythagorean theorem, square each of the legs, add these two squares, and set that
sum equal to the square of the hypotenuse. In equation form this is:
.
x%5E2+%2B+15%5E2+=+%282x%29%5E2
.
The square of 15 is 225 and %282x%29%5E2+=+4x%5E2. Substitute these into the equation and
it becomes:
.
x%5E2+%2B+225+=+4x%5E2
.
Subtract x%5E2 from both sides and the equation becomes:
.
225+=+3x%5E2
.
Divide both sides by 3 and it further reduces to:
.
75+=+x%5E2
.
Next take the square root of both sides and you get:
.
x+=+sqrt%2875%29+=+sqrt%2825%2A3%29+=+sqrt%2825%29%2Asqrt%283%29+=+5%2Asqrt%283%29
.
Notice the steps involved in simplifying the radical to get x+=+5%2Asqrt%283%29 as the
first possible answer.
.
But we said that the possibility exists that the hypotenuse is 15. In this case the two
legs are x and 2x. Applying the Pythagorean theorem to this problem results in:
.
x%5E2+%2B+%282x%29%5E2+=+15%5E2
.
Again substituting for %282x%29%5E2 and for 15%5E2 changes the equation to:
.
x%5E2+%2B+4x%5E2+=+225
.
Adding the two term on the left side results in:
.
5x%5E2+=+225
.
Divide both sides by 5 to get:
.
x%5E2+=+45
.
Then take the square root of both sides and you end up with:
.
x+=+sqrt%2845%29+=+sqrt%289%2A5%29+=+sqrt%289%29%2Asqrt%285%29+=+3%2Asqrt%285%29
.
This is the second possible answer x+=+3%2Asqrt%285%29
.
Which answer fits is dependent on which side is presumed to be the hypotenuse.
.
Hope this helps you to understand the problem and why the possibility exists for two answers.