Question

Three pieces of wood measure 20 cm, 41 cm and 44 cm. If the same amount is cut off each piece, the remaining lengths can be formed into a right triangle. What is the length that is cut off?

Answer 1

5 cm or 29 cm

Explanation:

Let x length is cut off from each measurements.

Thus, the new measurements would be: 20 - x, 41 - x, 44 - x, which forms the sides of a right triangle.

Here, (44 - x) will be hypotenuse and (20 - x) and (41 - x) will be perpendicular legs of the triangle.

Therefore, by Pythagoras theorem:

`{(41 - x)}^(2) + {(20 - x)}^(2) = {(44 - x)}^(2) \\ \\ \therefore \: {(41)}^(2) - 82x + {x}^(2) + {(20)}^(2) - 40x + {x}^(2) \\ = {(44)}^(2) - 88x + {x}^(2) \\ \\ \therefore \: 1681 - 122x + 2 {x}^(2) + 400 \\ = 1936 - 88x + {x}^(2) \\ \\ \therefore \: 2081 - 122x + {x}^(2) = 1936 - 88x \\ \\ \therefore \: {x}^(2) - 122x + 88x + 2081 - 1936 = 0 \\ \\ \therefore \: {x}^(2) - 34x + 145 = 0 \\ \\ \therefore \: {x}^(2) - 29x - 5x+ 145 = 0 \\ \\ \therefore \: x(x - 29) - 5(x - 29) = 0 \\ (x - 5)(x - 29) = 0 \\ x - 5 = 0 \: or \: x - 29 = 0 \\ \\ \therefore \: x = 5, \: 29`

So, the length that is cut off will be either 5 cm or 29 cm.