Question

1. use the given information about $\triangle abc$ with right angle $b$ to find the unknown side lengths. round your answer to the nearest hundredth. $c = 7$ and $m\angle a = 65^{\circ}$

a $a \approx 19.35, b \approx 5.47$
b $a \approx 11.22, b \approx 8.96$
c $a \approx 15.01, b \approx 16.56$
d $a \approx 9.45, b \approx 10.97$

Explanation:

Step1: Identify sides and angles

In right triangle \( \triangle ABC \) with right angle \( B \), \( c \) is the adjacent side to \( \angle A \), \( a \) is the opposite side to \( \angle A \), and \( b \) is the hypotenuse. We know \( c = 7 \), \( m\angle A = 65^\circ \).

Step2: Find side \( a \) (opposite to \( \angle A \))

We use the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). So \( \tan(65^\circ) = \frac{a}{c} \). Substituting \( c = 7 \), we get \( a = c \cdot \tan(65^\circ) \). \( \tan(65^\circ) \approx 2.1445 \), so \( a \approx 7 \times 2.1445 \approx 15.01 \).

Step3: Find side \( b \) (hypotenuse)

We use the cosine function: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). So \( \cos(65^\circ) = \frac{c}{b} \), which gives \( b = \frac{c}{\cos(65^\circ)} \). \( \cos(65^\circ) \approx 0.4226 \), so \( b \approx \frac{7}{0.4226} \approx 16.56 \).

Answer:

c. \( a \approx 15.01, b \approx 16.56 \)