Question 210200
a) Longer leg = x+7.
Hypotenuse = x+8.
b) {{{(x+8)^2 = x^2+(x+7)^2}}} Solve for x.
{{{x^2+16x+64 = x^2+x^2+14x+49}}}
{{{x^2+16x+64 = 2x^2+14x+49}}} Subtract {{{x^2}}} from both sides.
{{{16x+64 = x^2+14x+49}}} Subtract 16x from both sides.
{{{64 = x^2-2x+49}}} Subtract 64 from both sides.
{{{0 = x^2-2x-15}}}
{{{x^2-2x-15 = 0}}} Factor.
{{{(x+3)(x-5) = 0}}} so that...
{{{x = -3}}} or {{{highlight(x = 5)}}} Discard the negative solution as length is a positive quantity.
c) 
Hypotenuse is:
{{{c = x+8}}}
{{{c = 5+8}}}
{{{c = 13}}}
Longer side (a) is:
{{{a = x+7}}}
{{{a = 5+7}}}
{{{a = 12}}}
Shorter side (b) is:
{{{b = x}}} 
{{{b = 5}}} Check:
{{{c^2 = a^2+b^2}}}
{{{13^2 = 12^2+5^2}}}
{{{169 = 144+25}}}
{{{169 = 169}}} OK!