Question

a right triangle has one angle that measure 23°. the adjacent leg measures 27.6 cm and the hypotenuse measures 30 cm. what is the approximate area of the triangle? round to the nearest tenth. area of a triangle = \\(\frac{1}{2}bh\\) 68.7 cm² 161.8 cm² 381.3 cm² 450.0 cm²

Explanation:

Step1: Identify the formula and values

The formula for the area of a triangle is \( A = \frac{1}{2}bh \). Here, we can consider the two adjacent sides (let's say \( a = 27.6 \) cm and \( b = 30 \) cm) and the included angle \( \theta = 23^\circ \). The height \( h \) can be expressed as \( a\sin\theta \) (if we take \( b \) as the base). So the area formula becomes \( A=\frac{1}{2}ab\sin\theta \).

Step2: Substitute the values

Substitute \( a = 27.6 \), \( b = 30 \), and \( \theta = 23^\circ \) into the formula:
\( A=\frac{1}{2}\times27.6\times30\times\sin(23^\circ) \)

First, calculate \( \frac{1}{2}\times27.6\times30 = 13.8\times30 = 414 \)

Then, find \( \sin(23^\circ) \approx 0.3907 \)

Multiply: \( 414\times0.3907 \approx 414\times0.3907 \)

\( 400\times0.3907 = 156.28 \), \( 14\times0.3907 = 5.47 \), so total \( \approx 156.28 + 5.47 = 161.75 \approx 161.8 \)

Answer:

161.8 \( \text{cm}^2 \) (corresponding to the option "161.8 \( \text{cm}^2 \)")