1. The problem asks to find the length of the hypotenuse $h$ in two right-angled triangles given an angle and one side length. 2. We use the trigonometric functions sine, cosine, or tangent depending on the given sides and angles. The hypotenuse is related to the opposite or adjacent side by sine or cosine: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \quad \Rightarrow \quad h = \frac{\text{opposite}}{\sin(\theta)}$$ $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \quad \Rightarrow \quad h = \frac{\text{adjacent}}{\cos(\theta)}$$ 3. For triangle a): - Given angle $35^\circ$ and opposite side $3.0$ m. - Use sine formula: $$h = \frac{3.0}{\sin(35^\circ)}$$ Calculate $\sin(35^\circ) \approx 0.574$: $$h = \frac{3.0}{0.574}$$ Intermediate step with cancellation: $$h = \frac{3.0}{\cancel{0.574}} \Rightarrow h = 5.23$$ Rounded to nearest tenth: $$h \approx 5.2 \text{ m}$$ 4. For triangle b): - Given angle $39^\circ$ and adjacent side $5.0$ m. - Use cosine formula: $$h = \frac{5.0}{\cos(39^\circ)}$$ Calculate $\cos(39^\circ) \approx 0.777$: $$h = \frac{5.0}{0.777}$$ Intermediate step with cancellation: $$h = \frac{5.0}{\cancel{0.777}} \Rightarrow h = 6.44$$ Rounded to nearest tenth: $$h \approx 6.4 \text{ m}$$ 5. Summary: - Triangle a) hypotenuse $h \approx 5.2$ m - Triangle b) hypotenuse $h \approx 6.4$ m This method uses the sine or cosine ratio to find the hypotenuse when one side and an angle are known.