To find the area of the right triangle, we first need to determine the length of the opposite leg. Given:
- One of the angles is [tex]\(23^\circ\)[/tex],
- The adjacent leg measures [tex]\(27.6 \, \text{cm}\)[/tex],
- The hypotenuse measures [tex]\(30 \, \text{cm}\)[/tex].
Let's break down the solution step-by-step:
1. Determine the length of the opposite leg using the sine function:
- We know that [tex]\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)[/tex].
- Here, [tex]\(\theta = 23^\circ\)[/tex] and the hypotenuse is [tex]\(30 \, \text{cm}\)[/tex].
- The opposite leg can be found using the formula:
[tex]\[
\text{opposite leg} = \text{hypotenuse} \times \sin(23^\circ)
\][/tex]
2. Calculate the opposite leg:
- Using the sine value for [tex]\(23^\circ\)[/tex], the opposite leg is approximately [tex]\(11.7219 \, \text{cm}\)[/tex].
3. Calculate the area of the triangle:
- The formula for the area of a triangle is:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
- In our case, the adjacent leg [tex]\(27.6 \, \text{cm}\)[/tex] serves as the base and the calculated opposite leg [tex]\(11.7219 \, \text{cm}\)[/tex] serves as the height.
- Plugging these values in, we get:
[tex]\[
\text{Area} = \frac{1}{2} \times 27.6 \, \text{cm} \times 11.7219 \, \text{cm} \approx 161.7627 \, \text{cm}^2
\][/tex]
4. Round to the nearest tenth:
- The area [tex]\(161.7627 \, \text{cm}^2\)[/tex] rounded to the nearest tenth is [tex]\(161.8 \, \text{cm}^2\)[/tex].
Therefore, the approximate area of the triangle, rounded to the nearest tenth, is [tex]\(161.8 \, \text{cm}^2\)[/tex].
The correct answer is:
[tex]\[ 161.8 \, \text{cm}^2 \][/tex]
