Mohini wishes to fit three rods together in the shape of a right triangle. If the hypotenuse is 2 cm longer than the base and 4 cm longer than the shortest side, find the lengths of the rods.

Let the length of hypotenuse be x cm,

So, longer side = (x - 2) cm and shortest side = (x - 4) cm

Hypotenuse² = Sum of the squares of other two sides

∴ x² = (x - 2)² + (x - 4)²

$\Rightarrow x^2 = x^2 + 4 - 4x + x^2 + 16 - 8x \\[1em] \Rightarrow x^2 = 2x^2 - 12x + 20 \\[1em] \Rightarrow 2x^2 - x^2 - 12x + 20 = 0 \\[1em] \Rightarrow x^2 - 12x + 20 = 0 \\[1em] \Rightarrow x^2 - 10x - 2x + 20 = 0 \\[1em] \Rightarrow x(x - 10) - 2(x - 10) = 0 \\[1em] \Rightarrow (x - 2)(x - 10) = 0 \\[1em] \Rightarrow x - 2 = 0 \text{ or } x - 10 = 0 \\[1em] x = 2 \text{ or } x = 10$

x ≠ 2 as that will make shortest side negative (x - 4) = -2.

If x = 10 , x - 2 = 8 , x - 4 = 6.

Hence, the dimensions of triangle are Hypotenuse = 10 cm , Longer side = 8 cm , Shortest side = 6 cm.