a right triangle has one angle that measure $23^\\circ$. the adjacent leg measures 27.6 cm and the…

a right triangle has one angle that measure $23^\\circ$. the adjacent leg measures 27.6 cm and the hypotenuse measures 30 cm.\nwhat is the approximate area of the triangle? round to the nearest tenth.\narea of a triangle = $\\frac{1}{2}bh$\n$\\bigcirc$ $68.7\\ \\text{cm}^2$\n$\\bigcirc$ $161.8\\ \\text{cm}^2$\n$\\bigcirc$ $381.3\\ \\text{cm}^2$\n$\\bigcirc$ $450.0\\ \\text{cm}^2$

a right triangle has one angle that measure $23^\\circ$. the adjacent leg measures 27.6 cm and the hypotenuse measures 30 cm.\nwhat is the approximate area of the triangle? round to the nearest tenth.\narea of a triangle = $\\frac{1}{2}bh$\n$\\bigcirc$ $68.7\\ \\text{cm}^2$\n$\\bigcirc$ $161.8\\ \\text{cm}^2$\n$\\bigcirc$ $381.3\\ \\text{cm}^2$\n$\\bigcirc$ $450.0\\ \\text{cm}^2$

Answer

Explanation:

Step1: Find opposite leg length

We use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Rearranged, $\text{opposite} = \text{hypotenuse} \times \sin(\theta)$ $\text{opposite} = 30 \times \sin(23^\circ) \approx 30 \times 0.3907 = 11.721$ cm

Step2: Identify base and height

For the right triangle, the adjacent leg ($27.6$ cm) is the base, and the opposite leg ($11.721$ cm) is the height.

Step3: Calculate triangle area

Use the formula $A = \frac{1}{2}bh$ $A = \frac{1}{2} \times 27.6 \times 11.721$ $A = 13.8 \times 11.721 \approx 161.75$ cm²

Step4: Round to nearest tenth

$161.75 \approx 161.8$ cm²

Answer:

B. 161.8 cm²