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 base and height

In a right - triangle, we can take the adjacent leg as the base $b = 27.6$ cm. We need to find the height $h$. Using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 30$ cm and $b = 27.6$ cm. Let the height be $a$. Then $a=\sqrt{c^{2}-b^{2}}$.
$a=\sqrt{30^{2}-27.6^{2}}=\sqrt{(30 + 27.6)(30 - 27.6)}=\sqrt{57.6\times2.4}=\sqrt{138.24}\approx11.76$ cm.

Step2: Calculate the area

The formula for the area of a triangle is $A=\frac{1}{2}bh$. Substitute $b = 27.6$ cm and $h\approx11.76$ cm into the formula.
$A=\frac{1}{2}\times27.6\times11.76 = 13.8\times11.76=162.288\approx161.8$ $cm^{2}$.

Answer:

161.8 $cm^{2}$