"The shorter leg is 5 meters shorter than the longer leg", we can express this statement as follows
[tex]a=b-5[/tex]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
[tex]c=b+5[/tex]Where c is the hypotenuse and b is the longer leg.
We have to use Pythagorean's Theorem
[tex]c^2=a^2+b^2[/tex]First, let's combine the first equation with the last one
[tex]c^2=(b-5)^2+b^2[/tex]Then, we solve the square binomial
[tex]c^2=b^2-10b+25+b^2[/tex]Let's combine terms
[tex]c^2=2b^2-10b+25[/tex]Then, we combine the second equation with the last one above.
[tex](b+5)^2=2b^2-10b+25[/tex]Let's solve the square binomial.
[tex]b^2+10b+25=2b^2-10b+25[/tex]Now, we solve for b
[tex]\begin{gathered} 0=2b^2-b^2-10b-10b+25-25 \\ 0=b^2-20b \end{gathered}[/tex]Let's extract the greatest common factor b
[tex]b(b-20)=0[/tex]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.
[tex]c=b+5=20+5=25[/tex][tex]a=b-5=20-5=15[/tex]