To determine the area of the given right triangle, let's go through the solution step-by-step:
1. Identify the components of the triangle:
- Angle: [tex]\(23^\circ\)[/tex]
- Adjacent side: [tex]\(27.6 \, \text{cm}\)[/tex]
- Hypotenuse: [tex]\(30 \, \text{cm}\)[/tex]
2. Calculate the length of the opposite side:
- In a right triangle, the relationship between the sides can be described using the Pythagorean theorem:
[tex]\[
\text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2
\][/tex]
- Rearrange the equation to solve for the opposite side:
[tex]\[
\text{Opposite} = \sqrt{\text{Hypotenuse}^2 - \text{Adjacent}^2}
\][/tex]
- Substitute the given values:
[tex]\[
\text{Opposite} = \sqrt{30^2 - 27.6^2}
\][/tex]
- Compute the values under the square root:
[tex]\[
\text{Opposite} = \sqrt{900 - 761.76}
\][/tex]
[tex]\[
\text{Opposite} = \sqrt{138.24}
\][/tex]
[tex]\[
\text{Opposite} \approx 11.76 \, \text{cm}
\][/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]
- Here, the base is the adjacent side, and the height is the opposite side:
[tex]\[
\text{Area} = \frac{1}{2} \times 27.6 \, \text{cm} \times 11.76 \, \text{cm}
\][/tex]
- Calculate the value:
[tex]\[
\text{Area} = 0.5 \times 27.6 \times 11.76
\][/tex]
[tex]\[
\text{Area} \approx 162.3 \, \text{cm}^2
\][/tex]
4. Round the answer:
- The calculated area is approximately [tex]\(162.3 \, \text{cm}^2\)[/tex], rounded to the nearest tenth.
Among the given options, the correct one is:
[tex]\[ \boxed{161.8 \, \text{cm}^2} \][/tex]
