If the numeric perimeter of a right triangle is twice the numeric area of the triangle, and the hypotenuse is 9 more than one of the legs, what is the length of the longer leg?

Answer 1

Let x be the base of the triangle and y the height. Let the hypotenuse be 9+x. So now using the pythagorean theorem to find y it equals √(9+x)² -x² which gives √ 81 +18x. So now Perimeter = 2• Area and the perimeter

= 9 + 2x + y and the area = x•y/2 and so 9 +2x + y = x•y. Using the equation for y 9+2x + √81 +18x = x•√81 +18x. where √81 + 18x = 3√9 + 2x and so now 9 + 2x +3 √ 9 + 2x = 3x√9 + 2x simplifying 9 + 2x = 3(x-1)√9 +2x and now dividing both sides by the radical gives √9 + 2x = 3(x-1). To solve square both sides and get

9+ 2x = 9( x² -2x +1) and so 9 +2x = 9x² -18x +9 which gives 9x²-20x =0 the roots are x(9x-20)=0 and the one that applies is x= 20/9 and y = √81 + 18x which gives √81 + 40 = √121 = 11 where y= 11 is the length of the longer side.