Question 85290
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.)
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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:
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{{{x^2 + 15^2 = (2x)^2}}}
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The square of 15 is 225 and {{{(2x)^2 = 4x^2}}}. Substitute these into the equation and
it becomes:
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{{{x^2 + 225 = 4x^2}}}
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Subtract {{{x^2}}} from both sides and the equation becomes:
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{{{225 = 3x^2}}}
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Divide both sides by 3 and it further reduces to:
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{{{75 = x^2}}}
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Next take the square root of both sides and you get:
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{{{x = sqrt(75) = sqrt(25*3) = sqrt(25)*sqrt(3) = 5*sqrt(3)}}}
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Notice the steps involved in simplifying the radical to get {{{x = 5*sqrt(3)}}} 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:
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{{{x^2 + (2x)^2 = 15^2}}}
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Again substituting for {{{(2x)^2}}} and for {{{15^2}}} changes the equation to:
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{{{x^2 + 4x^2 = 225}}}
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Adding the two term on the left side results in:
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{{{5x^2 = 225}}}
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Divide both sides by 5 to get:
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{{{x^2 = 45}}}
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Then take the square root of both sides and you end up with:
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{{{x = sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5)}}}
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This is the second possible answer {{{x = 3*sqrt(5)}}}
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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.