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

The adjacent leg can be considered the base $b = 27.6$ cm. We need to find the height. 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 $a=\sqrt{c^{2}-b^{2}}$.

Step2: Calculate the height

$a=\sqrt{30^{2}-27.6^{2}}=\sqrt{(30 + 27.6)(30 - 27.6)}=\sqrt{57.6\times2.4}=\sqrt{138.24}=11.76$ cm.

Step3: Calculate the area

The area formula of a triangle is $A=\frac{1}{2}bh$. Substitute $b = 27.6$ cm and $h = 11.76$ cm. $A=\frac{1}{2}\times27.6\times11.76 = 161.7888\approx161.8$ $cm^{2}$.

Answer:

161.8 $cm^{2}$