Question: 14. Calculate the measure of $\angle BEC$ to the nearest degree. B E A C 4.6 m 3.7 m 39^\circ \angle \text{ at } C, \text{ with } EC \perp BA; \text{ triangle } BEC \text{ is on the left and triangle } ECA \text{ is on the right.

1. **State the problem:** We need to find the measure of $\angle BEC$ in the given figure where $EC$ is perpendicular to $BA$, $\angle C = 39^\circ$, and the sides $BE = 4.6$ m and $EA = 3.7$ m. 2. **Analyze the figure and given information:** Since $EC$ is perpendicular to $BA$, $\angle ECA = 90^\circ$. Triangle $ECA$ is right-angled at $C$. 3. **Use triangle $ECA$ to find $EC$:** Given $\angle C = 39^\circ$ and $EA = 3.7$ m, in right triangle $ECA$: $$\cos(39^\circ) = \frac{EC}{EA} \implies EC = EA \times \cos(39^\circ)$$ Calculate: $$EC = 3.7 \times \cos(39^\circ) \approx 3.7 \times 0.7771 = 2.8753 \text{ m}$$ 4. **Use triangle $BEC$ to find $\angle BEC$:** In triangle $BEC$, we know: - $BE = 4.6$ m - $EC \approx 2.8753$ m - $\angle ECB = 90^\circ$ (since $EC \perp BA$) We want $\angle BEC$, which is the angle at $E$ opposite side $BC$. 5. **Calculate $\angle BEC$ using tangent:** $$\tan(\angle BEC) = \frac{EC}{BE} = \frac{2.8753}{4.6} \approx 0.6249$$ 6. **Find $\angle BEC$:** $$\angle BEC = \tan^{-1}(0.6249) \approx 32^\circ$$ **Final answer:** $$\boxed{32^\circ}$$