1. Problem 4.1: In $\triangle DEF$, $\angle E = 90^\circ$, $DE = 15$ m, and $DF = 17$ m. Find the length of $EF$. 2. Use the Pythagorean theorem for right triangles: $$DF^2 = DE^2 + EF^2$$ 3. Substitute the known values: $$17^2 = 15^2 + EF^2$$ 4. Calculate squares: $$289 = 225 + EF^2$$ 5. Solve for $EF^2$: $$EF^2 = 289 - 225 = 64$$ 6. Take the square root: $$EF = \sqrt{64} = 8\text{ m}$$ --- 7. Problem 4.2: In $\triangle ABC$, $AB = 24$, $BC = 7$, and $AC = 25$. Show it is right-angled. 8. Check if Pythagorean theorem holds: $$AC^2 = AB^2 + BC^2$$ 9. Calculate squares: $$25^2 = 24^2 + 7^2$$ 10. Compute: $$625 = 576 + 49$$ 11. Sum right side: $$625 = 625$$ which is true, so $\triangle ABC$ is right-angled at $B$. --- 12. Problem 5.1: Calculate the area of a circle with diameter 20 cm. 13. Radius $r = \frac{20}{2} = 10$ cm. 14. Area formula: $$A = \pi r^2$$ 15. Substitute radius: $$A = \pi \times 10^2 = 100\pi\text{ cm}^2$$ --- 16. Problem 5.2: Calculate the perimeter of a shape made of a rectangle, a square, and an equilateral triangle with sides 8 cm and 3 cm. 17. The rectangle has length 8 cm and width 3 cm. 18. The square has side 3 cm. 19. The equilateral triangle has side 3 cm. 20. Perimeter is sum of outer edges. The shape's perimeter is: $$P = 8 + 3 + 3 + 3 + 3 + 3 = 23\text{ cm}$$ (Explanation: rectangle's length and width, plus three sides of the equilateral triangle and one side of the square that form the outer boundary.)