Question

find the area of this triangle. round to the nearest tenth.

? \text{ in.}^2

Explanation:

Identify the given values

We are given a triangle with:

• Side \(a = 12\text{ in.}\)
• Side \(b = 18\text{ in.}\)
• Included angle \(\theta = 147^\circ\)
• Third side \(c = 28.8\text{ in.}\) (not needed for the SAS area formula)

State the area formula

To find the area of a triangle when two sides and the included angle are known (SAS), we use the Sine Area Formula:

$$\text{Area} = \frac{1}{2} a b \sin(\theta)$$

Calculate the area

Substitute the given values into the formula:

$$\text{Area} = \frac{1}{2} \times 12 \times 18 \times \sin(147^\circ)$$

$$\text{Area} = 108 \times \sin(147^\circ)$$

Compute the final value

Using a calculator to find \(\sin(147^\circ)\):

$$\sin(147^\circ) \approx 0.544639$$

$$\text{Area} \approx 108 \times 0.544639 \approx 58.821$$

Round to the nearest tenth

Rounding \(58.821\) to the nearest tenth gives:

$$\text{Area} \approx 58.8\text{ in.}^2$$

Answer:

Find the area of this triangle. Round to the nearest tenth.

<blank>58.8</blank> \(\text{in.}^2\)