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.
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Let's first assume that the hypotenuse is 2x. That means that one leg is x and the other is
15.
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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:
.
.
The square of 15 is 225 and
. Substitute these into the equation and
it becomes:
.
.
Subtract
from both sides and the equation becomes:
.
.
Divide both sides by 3 and it further reduces to:
.
.
Next take the square root of both sides and you get:
.
.
Notice the steps involved in simplifying the radical to get
as the
first possible answer.
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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:
.
.
Again substituting for
and for
changes the equation to:
.
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Adding the two term on the left side results in:
.
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Divide both sides by 5 to get:
.
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Then take the square root of both sides and you end up with:
.
.
This is the second possible answer
.
Which answer fits is dependent on which side is presumed to be the hypotenuse.
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Hope this helps you to understand the problem and why the possibility exists for two answers.
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