1. **Calculate XY in triangle XYZ** Given: $XZ=6$ cm, $ZY=14$ cm, right angle at $Z$. Use Pythagoras theorem: $$XY=\sqrt{XZ^2+ZY^2}=\sqrt{6^2+14^2}=\sqrt{36+196}=\sqrt{232}=15.23\text{ cm (2 d.p.)}$$ 2. **Calculate hypotenuse in given right triangles** (a) Legs $6$ cm and $4$ cm: $$\text{Hypotenuse} = \sqrt{6^2+4^2}=\sqrt{36+16}=\sqrt{52}=7.21\text{ cm}$$ (b) Hypotenuse $20$ m, one leg $18$ m: Find other leg first: $$\text{Other leg} = \sqrt{20^2 - 18^2} = \sqrt{400 - 324} = \sqrt{76} = 8.72\text{ m}$$ 3. (a) **Find AB in triangle CBA** Given $CA=41$ cm (hypotenuse), $CB=40$ cm (base), right angle at $B$. AB is vertical leg: $$AB = \sqrt{CA^2 - CB^2} = \sqrt{41^2 - 40^2} = \sqrt{1681 - 1600} = \sqrt{81} = 9\text{ cm}$$ (b) **Find EF in triangle DEF** Given legs $DE=3.5$ m, $DF=8.4$ m, right angle at $D$. Hypotenuse EF: $$EF = \sqrt{3.5^2 + 8.4^2} = \sqrt{12.25 + 70.56} = \sqrt{82.81} = 9.10\text{ m}$$ 4. **Find $x$ in trapeziums S and T (similar shapes)** Given corresponding sides: $$\frac{x}{1.5} = \frac{4.5}{2.2} \implies x = 1.5 \times \frac{4.5}{2.2} = 3.07\text{ cm (2 d.p.)}$$ 5. **Triangles similarity to find $x$ and $y$** Given smaller triangle sides $8$, $6$, $y$, larger triangle sides $25$, $20$, $x$. Ratios: $$\frac{x}{8} = \frac{25}{8} = 3.125 \Rightarrow x=25$$ $$\frac{y}{6} = \frac{20}{6} = 3.333 \Rightarrow y = 6 \times \frac{20}{6} = 20$$ **Final answers:** 1. $XY=15.23$ cm 2a. Hypotenuse = 7.21 cm 2b. Other leg = 8.72 m 3a. $AB=9$ cm 3b. $EF=9.10$ m 4. $x=3.07$ cm 5a. $x=25$ cm 5b. $y=20$ cm