Question 709556: In a right triangle, the hypotenuse is 4 less than three
times the smaller side. The medium length side is one
more than the smallest side, and 1 less than the largest
side. What is the area of the triangle if the perimeter is
12?
Answer by josh_jordan(263)   (Show Source):
You can put this solution on YOUR website! To solve, let's first determine what we know:
1. The hypotenuse is 4 less than three times the smaller side.
2. The medium side is one more than the smallest side, and one less than the largest side.
3. The perimeter of the triangle is 12.
Let's translate these into a mathematical formula. Let A stand for the smallest side, B stand for the medium side, and C stand for the hypotenuse. We know the medium side is one more than the smallest side, and one less than the largest side (the hypotenuse). In other words, B = 1 + A and B = C - 1. We also know the hypotenuse is 4 less than 3 times the smaller side. In other words, C = 3A - 4.
We know the perimeter of the triangle is 12. The perimeter of a right triangle is A + B + C = P, where P stands for the perimeter.
If we replace B with 1 + A, and replace C with 3A - 4, we can solve for A:
A + 1 + A + 3A - 4 = 12
Combine like terms on the left side of the equal sign. This gives us
5A - 3 = 12
Add 3 to both sides of the equation, giving us
5A = 15
Finally, divide both sides by 5, which gives us A:
A = 3
Now that we know A, we can easily find B and C. We know that B is one more than A, so 1 + 3 = 4. So, B = 4. We know that C = 3A - 4, so C = 3(3) - 4, which means C = 5
Now that we know A = 3, B = 4, and C = 5, we can now find the area of our triangle. The formula to find the area of a right triangle, is (A x B)/2. So, replace A and B with their corresponding lengths to solve:
(3 x 4)/2 = 12/2 = 6
The area of the triangle is 6
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