Question

1. find the length of a.

triangle image with labels: side c = 5 cm, side b = 7 cm, angle at a = 30°; multiple choice options: 2.85 cm, 6.0 cm, 3.7 cm, 4.29 cm

Explanation:

Step1: Identify the Law to Use

We can use the Law of Cosines here. The Law of Cosines formula is \( a^{2}=b^{2}+c^{2}-2bc\cos(A) \), where \( b = 7\space\text{cm} \), \( c = 5\space\text{cm} \), and \( A=30^{\circ} \).

Step2: Substitute the Values

Substitute \( b = 7 \), \( c = 5 \), and \( \cos(30^{\circ})=\frac{\sqrt{3}}{2}\approx0.866 \) into the formula:
\[

$$\begin{align*} a^{2}&=7^{2}+5^{2}-2\times7\times5\times\cos(30^{\circ})\\ &=49 + 25-70\times0.866\\ &=74 - 60.62\\ &=13.38 \end{align*}$$

\]

Step3: Find the Length of \( a \)

Take the square root of \( a^{2} \) to find \( a \):
\( a=\sqrt{13.38}\approx3.66\approx3.7\space\text{cm} \)

Answer:

3.7 cm