Question 33 of 42intermediate🔧 ApplyNumerical3 marks

The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways.

Correct Answer

Way 1: Using Heron's Formula The sides of the triangle are a = 7 cm, b = 24 cm, c = 25 cm. Semi-perimeter, s = (7 + 24 + 25) / 2 = 56 / 2 = 28 cm. Area = √[s(s - a)(s - b)(s - c)] = √[28(28 - 7)(28 - 24)(28 - 25)] = √[28×2128 \times 21 × 4×34 \times 3] = 7056\sqrt{7056} = 84 cm².

Way 2: Using Base and Height By the Baudhāyana–Pythagoras theorem, 7² + 24² = 49 + 576 = 625 = 25². Thus, the triangle is right-angled. The two shorter sides, 7 cm and 24 cm, serve as the base and height. Area = (1/2) × base × height = (1/2) × 24×724 \times 7 = 84 cm².

Both methods give the same result. The area of the triangle is 84 cm².

Exercise: END-OF-CHAPTER EXERCISES | Q: 6 | (Page 32)
For More Understanding

Explanation

The question requires calculating the area using two different methods. The first method uses Heron's formula, which requires calculating the semi-perimeter. The second method uses the 'half base times height' formula, which is applicable because the given side lengths form a right-angled triangle, as verified by the Baudhāyana–Pythagoras theorem mentioned in the context.

Solution Steps

  1. Step 1: Calculate semi-perimeter for Heron's formula: s = (7 + 24 + 25) / 2 = 28 cm.

  2. Step 2: Apply Heron's formula: Area = √[28(28-7)(28-24)(28-25)] = √[28×2128 \times 21 × 4×34 \times 3] = 84 cm².

  3. Step 3: Verify the triangle is right-angled using Baudhāyana–Pythagoras theorem: 7² + 24² = 25².

  4. Step 4: Calculate area using base and height: Area = (1/2) × 24×724 \times 7 = 84 cm².

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