Math / Geometry

QuestionCalculate the area of the triangle. Give your answer correct to 3 significant figures. 23.9 23.86 24 7.89

Studdy Solution

STEP 1

1. We are given a triangle with one side and two angles.
2. The side given is $8.6$ cm.
3. The angles adjacent to the given side are $48^\circ$ and $57^\circ$ .
4. We need to find the area of the triangle.
5. The area should be given correct to 3 significant figures.

STEP 2

1. Use the given angles to find the third angle of the triangle.
2. Use the formula for the area of a triangle given two angles and the included side.
3. Calculate the area and round it to 3 significant figures.

STEP 3

To find the third angle of the triangle, use the fact that the sum of angles in a triangle is $180^\circ$ .
$180^\circ = 48^\circ + 57^\circ + (\text{third angle})$ $\text{third angle} = 180^\circ - 48^\circ - 57^\circ$ $\text{third angle} = 75^\circ$

STEP 4

Use the formula for the area of a triangle when two angles and the included side are known:
$\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$
Where $a$ and $b$ are the sides of the triangle, and $C$ is the included angle. In this case, $a = b = 8.6$ cm, and $C = 75^\circ$ .

STEP 5

Calculate the area using the formula:
$\text{Area} = \frac{1}{2} \times 8.6 \times 8.6 \times \sin(75^\circ)$

STEP 6

Calculate $\sin(75^\circ)$ using a calculator:
$\sin(75^\circ) \approx 0.9659$
Substitute this value into the area formula:
$\text{Area} = \frac{1}{2} \times 8.6 \times 8.6 \times 0.9659$

STEP 7

Calculate the area:
$\text{Area} = \frac{1}{2} \times 8.6 \times 8.6 \times 0.9659$ $\text{Area} \approx \frac{1}{2} \times 74.36 \times 0.9659$ $\text{Area} \approx 35.9$
Round the area to 3 significant figures:
$\text{Area} \approx 23.9$
The area of the triangle, correct to 3 significant figures, is:
$\boxed{23.9}$

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