Find the area of the triangle having the given measurements C=124 degrees, a=4 meters, b=6 meters
The Correct Answer and Explanation is:
To find the area of a triangle when you know two sides and the included angle, we can use the formula:Area=12absin(C)\text{Area} = \frac{1}{2}ab \sin(C)Area=21absin(C)
Where:
- a=4 ma = 4 \, \text{m}a=4m (side aaa)
- b=6 mb = 6 \, \text{m}b=6m (side bbb)
- C=124∘C = 124^\circC=124∘ (angle between sides aaa and bbb)
Now, we need to calculate the sine of 124∘124^\circ124∘. Using a calculator:sin(124∘)≈0.8387\sin(124^\circ) \approx 0.8387sin(124∘)≈0.8387
Substitute the values into the area formula:Area=12×4 m×6 m×0.8387\text{Area} = \frac{1}{2} \times 4 \, \text{m} \times 6 \, \text{m} \times 0.8387Area=21×4m×6m×0.8387Area=12×24 m2×0.8387\text{Area} = \frac{1}{2} \times 24 \, \text{m}^2 \times 0.8387Area=21×24m2×0.8387Area=12×0.8387 m2\text{Area} = 12 \times 0.8387 \, \text{m}^2Area=12×0.8387m2Area≈10.06 m2\text{Area} \approx 10.06 \, \text{m}^2Area≈10.06m2
So, the area of the triangle is approximately 10.06 square meters.
Explanation:
This formula relies on knowing two sides of a triangle and the angle between them. The reason it works is due to the relationship between the sides and the angle, which we take advantage of by using trigonometry. The sine of an angle in a triangle gives us a measure of the “height” relative to the base formed by the two known sides. By multiplying the length of the sides by the sine of the angle, we obtain a scaled height, which allows us to calculate the area effectively.
