SOLUTION: if a right triangle has a area 180cm^(2) and hypotenuse 41 cm, find the length of the two legs

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Question 218772: if a right triangle has a area 180cm^(2) and hypotenuse 41 cm, find the length of the two legs
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
If a right triangle has a area 180cm^(2) and hypotenuse 41 cm, find the length of the two legs.

Step 1. The Area A=ab%2F2=180 where a and b are legs of the triangle.

b=360%2Fa

Step 2. The Pythagorean Theorem is the sum of the squares of the legs (a and b) is equal to the square of the hypotenuse c

c%5E2=a%5E2%2Bb%5E2

c%5E2=a%5E2%2B%28360%2Fa%29%5E2

Step 3. Multiply by a^2 to both sides of the equation to get rid of denominator and c=41.

41%5E2%2Aa%5E2=a%5E2%2Aa%5E2%2B360%5E2

a%5E4-1681a%5E2%2B360%5E2=0

Step 4. Let x=a^2 to simplify equation as a quadratic

x%5E2-1681x%2B129600=0

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1681x%2B129600+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1681%29%5E2-4%2A1%2A129600=2307361.

Discriminant d=2307361 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1681%2B-sqrt%28+2307361+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1681%29%2Bsqrt%28+2307361+%29%29%2F2%5C1+=+1600
x%5B2%5D+=+%28-%28-1681%29-sqrt%28+2307361+%29%29%2F2%5C1+=+81

Quadratic expression 1x%5E2%2B-1681x%2B129600 can be factored:
1x%5E2%2B-1681x%2B129600+=+1%28x-1600%29%2A%28x-81%29
Again, the answer is: 1600, 81. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1681%2Ax%2B129600+%29



We have x=1600 and x=81, then a%5E2=1600 and a%5E2=81

Step 5. Taking the square root yields a=40 and a=9.

Then b is 360%2F40=9 or 360%2F9=40.

Step 6. ANSWER: The legs of the right triangle are 9 cm and 40 cm.

I hope the above steps and explanation were helpful.

For Step-By-Step videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry please visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Also, good luck in your studies and contact me at john@e-liteworks.com for your future math needs.

Respectfully,
Dr J

http://www.FreedomUniversity.TV