SOLUTION: A garden shaped like a right trianglr has one leg 6 meters less that the length of the other. The hypotenuse is 6 meters less than 2 times the length of the shorter leg. What are

Question 473138: A garden shaped like a right trianglr has one leg 6 meters less that the length of the other. The hypotenuse is 6 meters less than 2 times the length of the shorter leg. What are the lengths of the sides?
Answer by karaoz(32) (Show Source): You can put this solution on YOUR website!

There are three unknown values:
a = shorter leg
b = longer leg
c = hypotenuse

Translation of the relevant information into algebraic statements:
b = a + 6
c = 2a - 6
a = ?
b = ?

So, there are three variables and only two equations.
To be able to solve the system we need one more equation.
This equation will come from the fact that the triangle is a right triangle.
For any right triangle, it is true that a² + b² = c² and this will be our third equation.
Hence the system to solve is:

b = a + 6
c = 2a - 6
a² + b² = c²

We can quickly reduce the system by substituting b and c in the third equation by the expressions on the right hand sides of the first two equations.
This will give us:

a² + (a + 6)² = (2a - 6)²,
which is one equation with one unknown.

After squaring the expressions in brackets, we have
a² + a² + 12a + 36 = 4a² -24a + 36

After simplifying, the equation becomes:
2a² - 36a = 0, which is equivalent to:
2a(a - 18) = 0

There are two solutions to this equation: a = 0 and a = 18.
If a = 0 then b = 6 and c = -6, which is clearly not a real triangle even though the values satisfy all the three equations.
So, we will disregard this solution and keep only a = 18.

When a = 18 then b = 24 and c = 30.
So the answer to the question is:
The lengths of the sides are 18 meters and 24 meters.

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