The diagonal of a rectangle is 17 cm long and its perimeter is 46 cm. The area…

Concept

In a rectangle of length l and breadth b, three measurements are linked: the perimeter is 2(l + b), the diagonal d satisfies the Pythagoras relation l2 + b2 = d2, and the area is l × b.

The identity (l + b)2 = l2 + b2 + 2lb joins all three, so the product lb - the area - follows from the perimeter and the diagonal alone, without ever finding l and b separately.

Application

1. From the perimeter: 2(l + b) = 46 cm, so l + b = 23 cm.

2. From the diagonal: l2 + b2 = 172 = 289 cm2.

3. Square the side sum: (l + b)2 = 232 = 529, which expands to l2 + b2 + 2lb = 529.

4. Substitute the diagonal value: 289 + 2lb = 529, so 2lb = 240 and lb = 120.

5. The area is lb = 120 cm2.

Cross-check

The sides themselves are the roots of x2 - 23x + 120 = 0, that is x = 15 and x = 8. They reproduce every given value: perimeter 2(15 + 8) = 46 cm, diagonal √(152 + 82) = √289 = 17 cm, and area 15 × 8 = 120 cm2.