Question : One of the angles of a right-angled triangle is 15º, and the hypotenuse is 1 metre. The area of the triangle (in square cm) is:
Option 1: 1220
Option 2: 1200
Option 3: 1250
Option 4: 1215
Correct Answer: 1250
Solution :
The area of a right-angled triangle,
$\text{Area} = \frac{1}{2} \times\mathrm{base} \times\mathrm{height}$
Given that one of the angles is 15º.
$\mathrm{AB=AC \sin15^\circ}$
$\mathrm{BC=AC \cos15^\circ}$
⇒ $\text{Area} = \mathrm{\frac{1}{2} \times AC \sin15^\circ \times AC \cos15^\circ}$
⇒ $\text{Area} =\mathrm{ \frac{1}{4} \times AC^2 \times2 \times \sin( 15^\circ)\times \cos( 15^\circ)}$
⇒ $\text{Area} =\mathrm{ \frac{1}{4} \times (100 \; \text{cm})^2 \times \sin(30^\circ)}$
⇒ $\text{Area} =\mathrm{ \frac{1}{4} \times 10000 \; \text{sq. cm} \times \frac{1}{2}}$
⇒ $\text{Area} = 1250 \; \text{sq. cm}$
Hence, the correct answer is 1250.
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