Let me break down the part (b) of the question for you:

The problem revolves around finding lengths and angles in a triangle, specifically focusing on points X and Y on line segments, such that certain conditions are satisfied (like HX being horizontal). The calculation steps in the mark scheme seem to follow trigonometric rules and the Pythagorean theorem to solve for distances between points.

Explanation of Part (b):

1. Expression for [HX or YC]

   • The first part is calculating either the length of HX or YC: $[HX \text{ or } YC] = ("80" - 57.5) \cos 30$ or: $\sqrt{80^2 - 40^2} = 57.5 \cos 30$ This uses the cosine rule and Pythagoras' theorem to find the horizontal distance between H and X, or Y and C. The cosine part takes into account the angle 30 degrees.

     The value simplifies to: $19.4 \quad \text{or} \quad 85...$

2. Expression for [CX or YH]

   • The next step is finding the vertical distance for CX or YH: $[CX \text{ or } YH] = 57.5 \sin 30$ This part uses the sine rule to calculate the vertical component of the distance.

     Alternatively: $\frac{57.5 \times 40}{"80"} \quad \text{or} \quad 40 - ("80" - 57.5) \sin 30$ Simplified to: $28.75 \quad \text{or} \quad 28.8$

3. Expression for [DH]

   • Here, we calculate the distance DH: $[DH^2] = 100^2 + (19.485...)^2 + (28.75)^2$ These represent the distances squared and added, following the Pythagorean theorem. The mark scheme simplifies the sum of squares: $[= 11206.25]$ Resulting in: [ [DH] = \sqrt{11206.25} \quad \text{or} \quad 11206 \quad \text{or} \quad [DH] = 105.8... ]

4. Angle Calculation

   • Finally, the mark scheme computes the angle ∠EHD using the cosine rule: $\cos ∠EHD = \frac{102.5^2 + 11206.25 - 40^2}{2 \times 102.5 \times \sqrt{11206.25}}$ Simplified value: $[= 0.92679...]$

Key Takeaways:

• The calculations revolve around basic trigonometric formulas and the Pythagorean theorem to determine distances and angles.
• Values for lengths are calculated in stages, and trigonometric identities like sine and cosine are used based on given angles (like 30 degrees).

Would you like a detailed breakdown of any specific step or a deeper explanation of the trigonometric principles?

Follow-up Questions:

1. Do you want a step-by-step breakdown of how to derive [DH] using the Pythagorean theorem?
2. Should I clarify how the cosine rule was applied in this context?
3. Would you like a refresher on the sine and cosine rules used in this solution?
4. Should I explain how the specific angle (30 degrees) impacts the calculations of HX and YC?
5. Would you like me to elaborate on the final calculation of ∠EHD?

Tip:

When solving these types of problems, always ensure your triangle's angles and sides are labeled clearly to avoid confusion during the application of trigonometric rules.