Question 32 of 42intermediate🔧 ApplyNumerical3 marks

The sides of a triangle are in the ratio 2: 3: 4, and its perimeter is 45 cm. Find its area.

Correct Answer

75154 cm2\frac{75\sqrt{15}}{4} \text{ cm}^2

Exercise: END-OF-CHAPTER EXERCISES | Q: 5 | (Page 32)
For More Understanding

Explanation

To find the area of a triangle when the ratio of its sides and its perimeter are given, we first determine the actual side lengths. We do this by assigning a variable to the common multiplier in the ratio and using the perimeter to solve for it. Once the side lengths are known, we calculate the semi-perimeter. Finally, we apply Heron's formula, which is used in the textbook examples to find the area of triangles with known side lengths.

Solution Steps

  1. Find side lengths: Let the sides of the triangle be 2x2x, 3x3x, and 4x4x. The perimeter is 2x+3x+4x=9x=452x + 3x + 4x = 9x = 45 cm. Solving for xx, we get x=5x = 5 cm. Therefore, the sides are 1010 cm, 1515 cm, and 2020 cm.

  2. Calculate semi-perimeter: The semi-perimeter s=12(10+15+20)=452=22.5s = \frac{1}{2}(10 + 15 + 20) = \frac{45}{2} = 22.5 cm.

  3. Apply Heron's formula: Area = s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}.

  4. Compute terms: s−a=22.5−10=12.5s-a = 22.5 - 10 = 12.5 cm, s−b=22.5−15=7.5s-b = 22.5 - 15 = 7.5 cm, s−c=22.5−20=2.5s-c = 22.5 - 20 = 2.5 cm.

  5. Calculate area: Area = 22.5×12.5×7.5×2.5=452×252×152×52=843754=75154\sqrt{22.5 \times 12.5 \times 7.5 \times 2.5} = \sqrt{\frac{45}{2} \times \frac{25}{2} \times \frac{15}{2} \times \frac{5}{2}} = \frac{\sqrt{84375}}{4} = \frac{75\sqrt{15}}{4} cm2^2.

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