Question 1053258
let the sides and the hypotenuse of the triangle be:
{{{ a }}}, {{{ b }}}, and {{{ c }}} ( in feet )
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given:
(1) {{{ c = a + 1 }}}
(2) {{{ b + 1 = (1/3)*a }}}
(3) {{{ a^2 + b^2 = c^2 }}} ( pythagorean theorem )
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(2) {{{ b = (1/3)*a - 1 }}}
Substitute (1) and (2) into (3)
(3) {{{ a^2 + ((1/3)*a - 1 )^2 = ( a+1 )^2 }}}
(3) {{{ a^2 +  (1/9)*a^2 - (2/3)*a + 1  = a^2 + 2a + 1 }}}
(3) {{{ (1/9)*a^2 - (8/3)*a = 0  }}}
(3) {{{ a^2 - 24a = 0 }}}
(3) {{{ a*( a - 24 ) = 0 }}}
{{{ a = 24 }}}
and   
(1) {{{ c = a + 1 }}}
(1) {{{ c = 24 + 1 }}}
(1) {{{ c = 25 }}}
and
(2) {{{ b = (1/3)*a - 1 }}}
(2) {{{ b = (1/3)*24 - 1 }}}
(2) {{{ b = 7 }}}
The sides are 7, 24, and 25
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check answer:
(3) {{{ a^2 + b^2 = c^2 }}}
(3) {{{ 24^2 + 7^2 = 25^2 }}}
(3) {{{ 576 + 49 = 625 }}}
(3) {{{ 625 = 625 }}}
You can check (1) and (2)