The diagonal of a rectangular field is 17 metres and its perimeter is 46…

Concept

For a rectangle with length l, width w, diagonal d and perimeter P, the Pythagorean relation is d2 = l2 + w2, while P = 2(l + w).

The identity (l + w)2 = l2 + w2 + 2lw links these measurements to the area A = lw.

Application

1. From 2(l + w) = 46, we get l + w = 23 m.

2. From the diagonal, l2 + w2 = 172 = 289 m2.

3. Substitute these values into the identity: 232 = 289 + 2lw.

4. Thus, 529 = 289 + 2lw, so 2lw = 240 and lw = 120 m2.

Cross-check

1. The numbers whose sum is 23 and product is 120 are 15 and 8, so the field can have side lengths 15 m and 8 m.

2. These sides give perimeter 2(15 + 8) = 46 m and diagonal √(152 + 82) = √289 = 17 m, confirming the data independently.

Therefore, the area of the field is 120 m2.