The shorter leg of a right triangle is 7ft shorter than the longer leg. The hypotenuse is 7ft longer than the longer leg. Find the side lengths of the triangle

Hello!The answers are:[tex]Hypothenuse=28ft+7ft=35ft\\LongerLeg=28ft\\ShorterLeg=28ft-7ft=21ft[/tex]Why?Since we are working with a right triangle, we can use the Pythagorean Theorem, which states that:[tex]Hypothenuse^{2}=a^{2}+b^{2}[/tex]Then, we are given the following information:Let be "a" the shorter leg and "b" the the longer leg of the right triangle, so:[tex](7ft+b)^{2}=(b-7)^{2}+b^{2}[/tex]We can see that we need to perform the notable product, so:[tex](7ft+b)^{2}=(b-7ft)^{2}+b^{2}\\\\7ft*7ft+2*7ft*b+b^{2}=b^{2}-2*7ft*b+7ft*7ft+b^{2}\\\\49ft^{2} +14ft*b+b^{2}=b^{2}-14ft*b+49ft^{2}+b^{2}\\\\49ft^{2} +14ft*b+b^{2}=-14ft*b+49ft^{2}+2b^{2}\\\\-14ft*b+49ft^{2}+2b^{2}-(49ft^{2} +14ft*b+b^{2})=0\\\\-28ft*b+b^{2}=0\\\\b(-28ft+b)=0[/tex]We have that the obtained equation will be equal to 0 if: b is equal to 0 or b is equal to 28:[tex]0(-28+0)=0[/tex][tex]28(-28+28)=28(0)=0[/tex]So, since we are looking for the side of a leg, the result that we need its 28 feet.Hence, we have that the answers are:[tex]Hypothenuse=28ft+7ft=35ft\\LongerLeg=28ft\\ShorterLeg=28ft-7ft=21ft[/tex]Have a nice day!