1. **Problem statement:** We are given a children's slide with points X, Y, Z, U, V, and W. - |XY| = 10 m - |XV| = 15 m - \angle YXU = 15.2^\circ - \angle ZYW = 27^\circ We need to: (i) Show that |XU| = 9.65 m correct to 2 decimal places. (ii) Find |YZ| and then the overall length of the slide. 2. **Step (i): Find |XU|** - Triangle XYU has angle \angle YXU = 15.2^\circ at X. - |XY| = 10 m. - |XV| = 15 m is given but not directly needed here. Using the cosine rule in triangle XYU: $$|XU| = |XY| \cos(\angle YXU) = 10 \times \cos(15.2^\circ)$$ Calculate: $$\cos(15.2^\circ) \approx 0.9647$$ So: $$|XU| = 10 \times 0.9647 = 9.647 \approx 9.65 \text{ m}$$ 3. **Step (ii): Find |YZ| and overall length** - Given \angle ZYW = 27^\circ. - We assume right angles at U and W. Using sine rule or geometry, since |XY| = 10 m and |XU| = 9.65 m, the vertical drop |YU| is: $$|YU| = |XY| \sin(15.2^\circ) = 10 \times \sin(15.2^\circ)$$ Calculate: $$\sin(15.2^\circ) \approx 0.262$$ So: $$|YU| = 10 \times 0.262 = 2.62 \text{ m}$$ Next, find |YZ| using the angle 27^\circ and right triangle YZW: $$|YZ| = |YW| \cos(27^\circ)$$ But |YW| is unknown; however, since the slide continues from Y to Z to W, and W is right angle, we can use the geometry or given lengths. Assuming |YZ| is the horizontal component and |YW| the hypotenuse, and since the problem asks for |YZ| and overall length, we use the given data: Overall length = |XU| + |YZ| (slide length from X to U plus Y to Z) Since |YZ| is not directly given, we use the right triangle properties and given angles to find it. Assuming |YZ| = |YU| / tan(27^\circ): $$\tan(27^\circ) \approx 0.513$$ So: $$|YZ| = \frac{|YU|}{\tan(27^\circ)} = \frac{2.62}{0.513} = 5.11 \text{ m}$$ Finally, overall length: $$\text{Length} = |XU| + |YZ| = 9.65 + 5.11 = 14.76 \approx 15 \text{ m}$$ **Final answers:** - |XU| = 9.65 m - |YZ| = 5 m (nearest metre) - Overall length = 15 m (nearest metre)