Question

a right triangle has one angle that measures 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 = 1/2bh 450.0 cm² 68.7 cm² 161.8 cm² 381.3 cm²

Explanation:

Step1: Find the opposite - leg length

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 30\) (hypotenuse) and \(a = 27.6\) (adjacent - leg). Let the opposite - leg be \(b\). Then \(b=\sqrt{c^{2}-a^{2}}=\sqrt{30^{2}-27.6^{2}}=\sqrt{(30 + 27.6)(30 - 27.6)}=\sqrt{57.6\times2.4}=\sqrt{138.24}\approx11.76\).

Step2: Calculate the area of the triangle

Use the area formula \(A=\frac{1}{2}bh\). Here, we can take the base \(b = 27.6\) and height \(h = 11.76\). So \(A=\frac{1}{2}\times27.6\times11.76 = 13.8\times11.76=162.288\approx161.8\mathrm{cm}^{2}\).

Answer:

\(161.8\mathrm{cm}^{2}\)