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Let's start by analyzing the information and steps required to solve the two parts of the problem.

### Given Information:

- **Shape**: Right triangular prism \( ABCDEF \), with base \( ABCD \) on a horizontal plane.
- **Dimensions**:
  - \( AB = BC = DC = 100 \, \text{cm} \)
  - \( CF = DE = 40 \, \text{cm} \)
- **Angles**: \( \angle FBC = 30^\circ \)
- **Position of \( H \)**: Lies on line \( BF \), with \( BH = 57.5 \, \text{cm} \).
  
#### (a) Show that \( EH = 102.5 \, \text{cm} \)

This question asks to calculate the length of \( EH \) based on the information given.

### To approach this:

1. **Triangle \( BCF \)**:
   - We have \( BF = 100 \, \text{cm} \) and \( \angle FBC = 30^\circ \).
   - The height from \( C \) to \( BF \) (which forms a right triangle) can be calculated using trigonometry.

2. **Line \( EH \)**:
   - Once we find the length \( BF \), we can calculate the segment from \( B \) to \( H \), and then apply further geometry to find \( EH \).

I will begin by calculating \( EH \), and then proceed to solve part (b).

Let me calculate the value of \( EH \).The calculated length of \( EH \) is approximately \( 60.88 \, \text{cm} \). However, the problem states that \( EH \) should be \( 102.5 \, \text{cm} \), so it seems there's a need to reconsider the approach to ensure the result is aligned with the problem's conditions. Specifically, the horizontal component and use of the prism’s dimensions may need refining.

Let me adjust the method and check again for clarity.Let's review the situation and focus on the calculation method. In part (a), \( EH \) is the distance from point \( E \) to point \( H \), where \( H \) lies on line \( BF \). We need to calculate the proper diagonal between points considering both horizontal and vertical distances.

#### Steps:
- \( H \) is located on \( BF \), with \( BH = 57.5 \, \text{cm} \), and we need to connect it to \( E \), a point vertically above \( B \), by \( 40 \, \text{cm} \).
- We'll use the Pythagorean theorem in 3D space, considering the appropriate distances between these points.

Let me correct the approach now.After revising the calculation, the correct value for \( EH \) is approximately \( 70.04 \, \text{cm} \), not \( 102.5 \, \text{cm} \). This suggests that additional geometric considerations or a different interpretation of the figure might be necessary to resolve the discrepancy.

Would you like further assistance on this or prefer to move on to part (b), calculating \( \angle EHD \)?

Figure 1 shows a right triangular prism ABCDEF. The base ABCD is on a horizontal plane. CF = DE = 40 cm, EF = DC = AB = 100 cm, and angle FBC = 30 degrees. H lies on line BF with BH = 57.5 cm. (a) Show that EH = 102.5 cm. (b) Calculate angle EHD, to 3 significant figures.

This problem explores a right triangular prism, requiring the use of trigonometry and 3D geometry. Learn how to calculate the distance EH = 102.5 cm and angle EHD in degrees using the Pythagorean Theorem and Cosine Rule.

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