The hypotenuse of a grassy land in the shape of a right triangle is 1 metre more than twice the shortest side. If the third side is 7 metres more than the shortest side, find the sides of the grassy land.

Let the shortest side be x metres

So, hypotenuse = (2x + 1) metres and third side = (x + 7) metres

Hypotenuse² = Perpendicular² + Base²

∴ (2x + 1)² = x² + (x + 7)²

$\Rightarrow 4x^2 + 1 + 4x = x^2 + x^2 + 49 + 14x \\[1em] \Rightarrow 4x^2 + 1 + 4x = 2x^2 + 49 + 14x \\[1em] \Rightarrow 4x^2 - 2x^2 + 1 - 49 + 4x - 14x = 0 \\[1em] \Rightarrow 2x^2 - 48 - 10x = 0 \\[1em] \Rightarrow 2(x^2 - 24 - 5x) = 0 \\[1em] \Rightarrow x^2 - 5x - 24 = 0 \\[1em] \Rightarrow x^2 - 8x + 3x - 24 = 0 \\[1em] \Rightarrow x(x - 8) + 3(x - 8) = 0 \\[1em] \Rightarrow (x + 3)(x - 8) = 0 \\[1em] \Rightarrow x + 3 = 0 \text{ or } x - 8 = 0 \\[1em] x = -3 \text{ or } x = 8$

Since no side of a triangle can be negative hence, x ≠ -3

If x = 8 , (2x + 1) = 17 , (x + 7) = 15

Hence, the hypotenuse of the triangle is 17 metres while the shorter sides are 8 metres and 15 metres.