Question: EXERCISE
1. Find b.
14 cm
34°
77 cm
B
A
?
C
Triangle B A C with top side B-A labeled 14 cm, left side B-C labeled 77 cm, angle at B labeled 34°, unknown side A-C marked with ?; position_hint: center
1. **State the problem:** We need to find the length of side $b = AC$ in triangle $BAC$ where $BA = 14$ cm, $BC = 77$ cm, and angle $B = 34^\circ$.
2. **Identify the formula:** Use the Law of Cosines since we know two sides and the included angle:
$$b^2 = a^2 + c^2 - 2ac \cos B$$
where $a = BA = 14$ cm, $c = BC = 77$ cm, and $B = 34^\circ$.
3. **Substitute known values:**
$$b^2 = 14^2 + 77^2 - 2 \times 14 \times 77 \times \cos 34^\circ$$
4. **Calculate each term:**
$$14^2 = 196$$
$$77^2 = 5929$$
$$2 \times 14 \times 77 = 2156$$
$$\cos 34^\circ \approx 0.8290$$
5. **Evaluate the expression:**
$$b^2 = 196 + 5929 - 2156 \times 0.8290$$
$$b^2 = 6125 - 1787.524$$
$$b^2 = 4337.476$$
6. **Find $b$ by taking the square root:**
$$b = \sqrt{4337.476} \approx 65.87$$
7. **Final answer:**
The length of side $b = AC$ is approximately **65.87 cm**.