Question

the shorter leg of a right triangle is 5 meters shorter than the longer leg the hypotenuse is 5m longer than the longer leg find the side length of the triangle length of the shorter leg: __mlengths of the longer leg:__m length of the hypotenuse: __m

Answer 1

"The shorter leg is 5 meters shorter than the longer leg", we can express this statement as follows

`a=b-5`

Where a is the shorter leg and b is the longer leg.

"The hypotenuse is 5m longer than the longer leg", we can express this statement as follows

`c=b+5`

Where c is the hypotenuse and b is the longer leg.

We have to use Pythagorean's Theorem

`c^2=a^2+b^2`

First, let's combine the first equation with the last one

`c^2=(b-5)^2+b^2`

Then, we solve the square binomial

`c^2=b^2-10b+25+b^2`

Let's combine terms

`c^2=2b^2-10b+25`

Then, we combine the second equation with the last one above.

`(b+5)^2=2b^2-10b+25`

Let's solve the square binomial.

`b^2+10b+25=2b^2-10b+25`

Now, we solve for b

`\begin{gathered} 0=2b^2-b^2-10b-10b+25-25 \\ 0=b^2-20b \end{gathered}`

Let's extract the greatest common factor b

`b(b-20)=0`

This means b = 0 and b = 20, where the solution that makes sense to the problem is b = 20.

Now, we find the other sides.

`c=b+5=20+5=25`
`a=b-5=20-5=15`

Hence, the shorter leg is 15m, the longer leg is 20m, and the length of the hypotenuse is 25m.