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: Find the opposite side length

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 30\) (hypotenuse) and \(a=27.6\) (adjacent side). Let the opposite side be \(b\).

$$b=\sqrt{c^{2}-a^{2}}=\sqrt{30^{2}-27.6^{2}}=\sqrt{(30 + 27.6)(30 - 27.6)}=\sqrt{57.6\times2.4}=\sqrt{138.24}\approx11.76$$

Alternatively, using trigonometry \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). But since we know two sides of the right - triangle (using Pythagorean theorem is straightforward here).

Step2: Calculate the area

Using the formula \(A=\frac{1}{2}bh\). Here, \(b = 27.6\) and \(h\approx11.76\)

$$A=\frac{1}{2}\times27.6\times11.76 = 13.8\times11.76=161.288\approx161.8$$

Answer:

B. \(161.8\ cm^{2}\)