1. **Problem:** Triangle ABC is an enlargement of triangle PQR. Given sides of PQR are 7 cm, 5 cm, and 8 cm, and sides of ABC are 12 cm, 5 cm, and unknown. Calculate the surface area of triangle ABC. 2. **Formula and rules:** The surface area of a triangle is given by $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$. For similar triangles, the ratio of areas is the square of the scale factor. 3. **Find scale factor:** Compare corresponding sides. Given PQR side 7 cm corresponds to ABC side 12 cm. $$\text{Scale factor} = \frac{12}{7}$$ 4. **Calculate area of PQR:** Use Heron's formula. Semi-perimeter $$s = \frac{7 + 5 + 8}{2} = 10$$ Area $$= \sqrt{s(s-7)(s-5)(s-8)} = \sqrt{10 \times 3 \times 5 \times 2} = \sqrt{300} = 10\sqrt{3}$$ cm$^2$ 5. **Calculate area of ABC:** Area scales by square of scale factor. $$\text{Area}_{ABC} = \left(\frac{12}{7}\right)^2 \times 10\sqrt{3} = \frac{144}{49} \times 10\sqrt{3} = \frac{1440\sqrt{3}}{49} \approx 50.7$$ cm$^2$ **Final answer:** Surface area of triangle ABC is approximately 50.7 cm$^2$.