SOLUTION: The sum of the lengths of one side of a right triangle and the hypotenuse is 20cm. The other leg is 5cm shorter than the hypotenuse. Find the length of each side. x^2+(x-5)^2=2

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Question 122779: The sum of the lengths of one side of a right triangle and the hypotenuse is 20cm. The other leg is 5cm shorter than the hypotenuse. Find the length of each side.
x^2+(x-5)^2=20^2
x^2+x^2-25x+25=400
2x^2-25x-375=0
x^2-5x-178.50=0
I just want to double check my steps to make sure I did this right. Thanks is advance.
Found 2 solutions by GSems22, bucky:
Answer by GSems22(3) About Me  (Show Source):
You can put this solution on YOUR website!
Your answer appears correct in my eyes. That would be how I would solve it. Good Job!

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
The way I solved this problem was to start with three equations.
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Start by letting x and y equal the two legs of the triangle and h equal the hypotenuse.
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The first sentence in the problem says the sum of one side and the hypotenuse equals 20 cm.
In equation form this can be written as:
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x+%2B+h+=+20
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Solve this equation for x and you have:
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x+=+20+-+h
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The next sentence in the problem says the other leg of the triangle is 5 cm less than the
hypotenuse. In equation form this is:
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y+=+h+-+5
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The final equation in the series comes from the Pythagorean theorem which for this problem
says that the sum of the squares of the legs equals the square of the hypotenuse. In
this problem it is:
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x%5E2+%2B+y%5E2+=+h%5E2
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Substituting 20 - h for x and h - 5 for y results in:
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%2820+-+h%29%5E2+%2B+%28h+-+5%29%5E2+=+h%5E2
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Squaring out the two terms on the left side gives:
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400+-+40h+%2B+h%5E2+%2B+h%5E2+-+10h+%2B+25+=+h%5E2
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Combining the like terms on the left side results in:
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2h%5E2+-+50h+%2B+425+=+h%5E2
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Subtract h^2 from both sides and you get the standard quadratic form of:
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h%5E2+-+50h+%2B+425+=+0
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If you apply the quadratic formula to get the solutions to this problem, you start with:
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h+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
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and substitute 1 for a, -50 for b, and 425 for c to get:
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h+=+%28-%28-50%29+%2B-+sqrt%28+%28-50%29%5E2-4%2A1%2A425+%29%29%2F%282%2A1%29+
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In the radical %28-50%29%5E2+-4%2A1%2A425+=+2500+-+1700+=+800 and substituting this gives:
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h+=+%28-%28-50%29+%2B-+sqrt%28+800%29%29%2F%282%2A1%29+
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The -(-50) = +50. The square root of 800 = 28.28427125. And the denominator is just 2. These
simplifications result in:
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h+=+%2850+%2B-+28.28427125%29%2F2
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Dividing the two terms in the numerator by the denominator results in:
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h+=+25+%2B-+14.14213562
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This means that there are two possible answers for h. The first is 25 + 14.14213562
which is 39.14213562. However, this cannot be the answer because we know that x + h must
be 20. But if h is 39.14213562 then x has to be a negative number if the sum of x and h
is to be 20.
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So the next trial is to assume the answer is h = 25 - 14.14213562 and this subtraction leads
to:
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h = 10.85786438 cm
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Then we can use the equation x + h = 20 to find x. Substitute 10.85786438 for h and x becomes
defined by the equation:
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x+%2B+10.85786438+=+20
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Subtract 10.85786438 from both sides of this equation and the result is:
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x+=+20+-+10.85786438+=+9.14213562
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So one leg is 9.14213562 cm.
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The other leg of the triangle (side y) is 5 cm smaller than h and this is:
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y+=+h+-+5+=+10.85786438+-+5+=+5.85786438
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So the length of the final side is 5.85786438 cm.
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In summary, the three sides are: hypotenuse = 10.85786438 cm, leg x = 9.14213562 cm, and
leg y = 5.85786438 cm.
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If you check these values out you will find that they satisfy all three equations.
In other words:
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x + h = 9.14213562 + 10.85786438 = 20
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and:
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y = h - 5
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5.85786432 = 10.85786438 - 5
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and
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x^2 + y^2 = h^2
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(9.14213562)^2 + (5.85786432)^2 = (10.85786438)^2
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Hope this helps you to see how to set up and solve this problem.
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