The longer leg of a right triangle is 1 m longer than the shorter leg. The hypotenuse is 9 m longer than the shorter leg Find the side lengths of the triangl

Let [tex]x[/tex] be the length of the shorter leg.The other leg is 1m longer, so its length is [tex]x+1[/tex]The hypothenuse is 9m longer, so its length is [tex]x+9[/tex]The pythagorean theorem states that the sum of the squares of the legs is the square of the hypothenuse, so we have[tex]x^2+(x+1)^2=(x+9)^2[/tex]Expanding the squares gives[tex]x^2+x^2+2x+1=x^2+18x+81[/tex]Move all to the left hand side:[tex]x^2-16x-80=0[/tex]This equation has solutions [tex]x=-4[/tex] and [tex]x=20[/tex]We can't accept the first solution, because it would lead to the side lengths[tex]x=-4,\quad x+1=-3,\quad x+9=5[/tex]And we can't have negative side lengths.The other solution is fine, because it leads to the side lengths[tex]x=20,\quad x+1=21,\quad x+9=29[/tex]So, the side lengths are 20 (shorter leg), 21 (longer leg), 29 (hypothenuse)