Question : The three sides of a triangle are 7 cm, 9 cm, and 8 cm. What is the area of the triangle?
Option 1: $12 \sqrt{3} \;\mathrm{Sq} . \mathrm{cm}$
Option 2: $10\sqrt{3} \;\mathrm{Sq} . \mathrm{cm}$
Option 3: $12 \sqrt{5} \;\mathrm{Sq} . \mathrm{cm}$
Option 4: $2 \sqrt{5} \;\mathrm{Sq} . \mathrm{cm}$
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Correct Answer: $12 \sqrt{5} \;\mathrm{Sq} . \mathrm{cm}$
Solution :
Using the formula:
$s = \frac{(a + b + c)}{2}$
$s = \frac{(7 + 9 + 8)}{2}$
$s = \frac{24}{2}$
$s = 12$ cm
Now, plug the values of a, b, c, and s into Heron's formula to find the area of the triangle:
$\sqrt{(s(s - a)(s - b)(s - c))}$
$=\sqrt{(12(12 - 7)(12 - 9)(12 - 8))}$
$=\sqrt{(12 × 5 × 3 × 4)}$
$=12\sqrt{5}$ Sq. cm
Therefore, the area of the triangle with side lengths 7 cm, 9 cm, and 8 cm is approximately $12\sqrt{5}$ Sq. cm
Hence, the correct answer is $12 \sqrt{5} \;\mathrm{Sq}. \mathrm{cm}$.
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