To solve this problem, let's go through each part of the question:

Part (a): Finding the Length of $BC$

From the diagram, we can observe that triangle $CFB$ is a right-angled triangle with:

• $CF = 15$ cm (the base)
• $FB = 17$ cm (the height)

To find the length of $BC$, we can use the Pythagorean theorem: $BC = \sqrt{CF^2 + FB^2}$ Substituting the values: $BC = \sqrt{15^2 + 17^2} = \sqrt{225 + 289} = \sqrt{514}$ Calculating this gives: $BC \approx 22.7 \, \text{cm} \quad \text{(to 1 decimal place)}$

Part (b): Finding the Length of $BD$

Now that we know $BC \approx 22.7$ cm, we move on to finding $BD$ in triangle $BCD$. In the triangular prism, $BD$ forms the hypotenuse of the right triangle $BCD$, where:

• $BC \approx 22.7$ cm (calculated from part (a))
• $CD = 25$ cm (as given in the diagram)

Using the Pythagorean theorem again: $BD = \sqrt{BC^2 + CD^2}$ Substituting the values: $BD = \sqrt{(22.7)^2 + 25^2} = \sqrt{515.29 + 625} = \sqrt{1140.29}$ Calculating this gives: $BD \approx 33.8 \, \text{cm} \quad \text{(to 1 decimal place)}$

Summary of Answers:

• Length of $BC$ ≈ 22.7 cm
• Length of $BD$ ≈ 33.8 cm

Would you like a detailed breakdown of each calculation step?

Here are 5 related questions that may expand your understanding:

1. How can the Pythagorean theorem be used in three-dimensional geometry problems?
2. What are the implications of the Pythagorean theorem in different shapes and prisms?
3. How can we verify our answers in right-angled triangular prism problems?
4. Can you determine the volume of this triangular prism with the given lengths?
5. How would the length calculations change if the prism were not right-angled?

Tip: Always double-check your triangle sides to ensure they form a right angle before applying the Pythagorean theorem.