1. Problem 14: Calculate the area of a triangle with sides 17.1 cm, 22.8 cm, and 28.5 cm. 2. We use Heron's formula for the area of a triangle when all three sides are known: $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter. 3. Calculate the semi-perimeter: $$s = \frac{17.1 + 22.8 + 28.5}{2} = \frac{68.4}{2} = 34.2$$ 4. Substitute values into Heron's formula: $$A = \sqrt{34.2(34.2-17.1)(34.2-22.8)(34.2-28.5)}$$ $$= \sqrt{34.2 \times 17.1 \times 11.4 \times 5.7}$$ 5. Calculate the product inside the square root: $$34.2 \times 17.1 = 584.82$$ $$584.82 \times 11.4 = 6667.548$$ $$6667.548 \times 5.7 = 38002.1236$$ 6. Find the square root: $$A = \sqrt{38002.1236} \approx 194.9$$ 7. Final answer for problem 14: The area of the triangle is approximately $194.9$ cm$^2$ correct to one decimal place. --- 1. Problem 17: Calculate the area of a scalene triangle with sides 12.9 cm, 17.2 cm, and 21.5 cm. 2. Use Heron's formula again: $$s = \frac{12.9 + 17.2 + 21.5}{2} = \frac{51.6}{2} = 25.8$$ 3. Substitute into the formula: $$A = \sqrt{25.8(25.8-12.9)(25.8-17.2)(25.8-21.5)}$$ $$= \sqrt{25.8 \times 12.9 \times 8.6 \times 4.3}$$ 4. Calculate the product: $$25.8 \times 12.9 = 332.82$$ $$332.82 \times 8.6 = 2866.252$$ $$2866.252 \times 4.3 = 12319.8846$$ 5. Square root: $$A = \sqrt{12319.8846} \approx 111.0$$ 6. Final answer for problem 17: The area of the triangle is approximately $111.0$ cm$^2$ correct to one decimal place. --- 1. Problem 19: Calculate the area of a kite with diagonals 12.5 cm and 18.3 cm. 2. The formula for the area of a kite is: $$A = \frac{1}{2} d_1 d_2$$ where $d_1$ and $d_2$ are the lengths of the diagonals. 3. Substitute the values: $$A = \frac{1}{2} \times 12.5 \times 18.3$$ 4. Calculate: $$A = \frac{1}{2} \times 228.75 = 114.375$$ 5. Final answer for problem 19: The area of the kite is approximately $114.4$ cm$^2$ correct to one decimal place. --- 1. Problem 22: Calculate the area of the kite shown with diagonals 19.4 mm + 67.9 mm (vertical) and 29.1 mm (horizontal segments each side, total diagonal $2 \times 29.1 = 58.2$ mm). 2. The total vertical diagonal length is: $$d_1 = 19.4 + 67.9 = 87.3 \text{ mm}$$ 3. The horizontal diagonal length is: $$d_2 = 29.1 + 29.1 = 58.2 \text{ mm}$$ 4. Use the kite area formula: $$A = \frac{1}{2} d_1 d_2 = \frac{1}{2} \times 87.3 \times 58.2$$ 5. Calculate: $$A = \frac{1}{2} \times 5083.86 = 2541.93$$ 6. Final answer for problem 22: The area of the kite is approximately $2541.9$ mm$^2$ correct to one decimal place.