Question

solve the right triangle shown in the figure. a = 12.4, b = 22.6 a. what is the length of the side? c ≈ 25.78 (round to the nearest hundredth as needed.) b. what are the angles? a ≈ 28.8° (round to the nearest tenth as needed.) b ≈ 61.2° (round to the nearest tenth as needed.) c = °

Explanation:

Step1: Recall right - triangle properties

In a right - triangle, $\angle C = 90^{\circ}$ by definition.

Step2: Calculate side length c using Pythagorean theorem

$c=\sqrt{a^{2}+b^{2}}=\sqrt{12.4^{2}+22.6^{2}}=\sqrt{153.76 + 510.76}=\sqrt{664.52}\approx25.78$.

Step3: Find angle A using tangent function

$\tan A=\frac{a}{b}=\frac{12.4}{22.6}$, so $A=\arctan(\frac{12.4}{22.6})\approx28.8^{\circ}$.

Step4: Find angle B

Since the sum of angles in a triangle is $180^{\circ}$, and $\angle C = 90^{\circ}$, $\angle A+\angle B+\angle C = 180^{\circ}$. Then $B = 90^{\circ}-A=90^{\circ}-28.8^{\circ}=61.2^{\circ}$.

Answer:

a. $c\approx25.78$
b. $A\approx28.8^{\circ}$, $B\approx61.2^{\circ}$, $C = 90^{\circ}$