1. **State the problem:** We have a composite figure made of two triangles sharing a side of 15 cm. The left triangle has sides 15 cm and 25 cm with an included angle of 50°. The right triangle shares the 15 cm side and has an included angle of 70°. The base of the entire figure is 30 cm. We want the total area inside the frame to be at least 800 cm². 2. **Formulas and rules:** - Area of a triangle using two sides and included angle: $$\text{Area} = \frac{1}{2}ab\sin(C)$$ - Law of Cosines: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ - Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 3. **Calculate area of left triangle:** Given sides $a=15$ cm, $b=25$ cm, and included angle $C=50^\circ$, $$\text{Area}_\text{left} = \frac{1}{2} \times 15 \times 25 \times \sin(50^\circ)$$ Calculate $\sin(50^\circ) \approx 0.7660$, $$\text{Area}_\text{left} = \frac{1}{2} \times 15 \times 25 \times 0.7660 = 143.625\text{ cm}^2$$ 4. **Find the side opposite to 50° in left triangle (shared side):** Using Law of Cosines, $$c^2 = 15^2 + 25^2 - 2 \times 15 \times 25 \times \cos(50^\circ)$$ Calculate $\cos(50^\circ) \approx 0.6428$, $$c^2 = 225 + 625 - 750 \times 0.6428 = 850 - 482.1 = 367.9$$ $$c = \sqrt{367.9} \approx 19.18\text{ cm}$$ This side $c$ is the shared side between the two triangles. 5. **Calculate area of right triangle:** Given one side is 15 cm (shared side), and included angle is $70^\circ$, but we need the other side adjacent to this angle to use the area formula. 6. **Find the other side of right triangle using Law of Cosines:** The base of the entire figure is 30 cm, so the other side of the right triangle is $30 - 19.18 = 10.82$ cm. 7. **Calculate area of right triangle:** Using sides $a=15$ cm, $b=10.82$ cm, and included angle $70^\circ$, $$\text{Area}_\text{right} = \frac{1}{2} \times 15 \times 10.82 \times \sin(70^\circ)$$ Calculate $\sin(70^\circ) \approx 0.9397$, $$\text{Area}_\text{right} = \frac{1}{2} \times 15 \times 10.82 \times 0.9397 = 76.3\text{ cm}^2$$ 8. **Calculate total area:** $$\text{Area}_\text{total} = 143.625 + 76.3 = 219.925\text{ cm}^2$$ 9. **Check if total area meets requirement:** The total area $219.925$ cm² is less than the required $800$ cm². **Final answer:** The current configuration does not meet the requirement of at least 800 cm² area inside the frame.