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The diagram shows triangle ABC - HSC - SSCE Mathematics Standard - Question 35 - 2023 - Paper 1

Question 35

The diagram shows triangle ABC. Calculate the area of the triangle, to the nearest square metre.

Worked Solution & Example Answer:The diagram shows triangle ABC - HSC - SSCE Mathematics Standard - Question 35 - 2023 - Paper 1

Step 1

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Answer

To find the side x opposite to angle C, we can use the sine rule:

$\frac{x}{\sin 60^\circ} = \frac{12}{\sin 25^\circ}$

Rearranging gives:

$x = 12 \times \frac{\sin 60^\circ}{\sin 25^\circ}$

Calculating this:

$x \approx 12 \times 2.440 \approx 24.59$

Step 2

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Answer

The formula for the area of a triangle is given by:

$\text{Area} = \frac{1}{2}ab \sin C$

In this case, where a = 12 m, b = x (24.59 m), and angle C = 95°:

$\text{Area} = \frac{1}{2} \times 12 \times 24.59 \times \sin 95^\circ$

Calculating gives:

$\text{Area} \approx \frac{1}{2} \times 12 \times 24.59 \times 1 \approx 146.98 \text{ m}^2$

Thus, rounding to the nearest square metre, the area is approximately 147 m².

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