An umbrella is made by stitching 10 congruent triangular pieces of cloth in…

Concept

For a triangle with side lengths a, b, and c, Heron’s formula gives its area as √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 is the semi-perimeter.

When congruent pieces are divided equally between colours, the cloth required for one colour equals the area of one piece multiplied by the number of pieces of that colour.

Application

1. For one triangular piece, the semi-perimeter is s = (20 + 50 + 50)/2 = 60 cm.

2. The three differences are s − 20 = 40 cm, s − 50 = 10 cm, and s − 50 = 10 cm.

3. By Heron’s formula, the area of one piece is √(60 × 40 × 10 × 10) = √240000 = 200√6 cm2.

4. There are 10 pieces in two colours, with 5 congruent pieces of each colour.

5. Therefore, cloth required for one colour is 5 × 200√6 = 1000√6 cm2.

Cross-check

The triangle is isosceles. Its height is √(502 − 102) = 20√6 cm, so its area is (1/2) × 20 × 20√6 = 200√6 cm2, confirming the Heron’s-formula result.

Hence, 1000√6 cm2 of cloth of each colour is required.