To solve the problem, we’ll go through each part step-by-step.

Part (a): Finding the length of $BC$ in the right-angled triangular prism

From the diagram, we see a triangle $\triangle CEF$ where:

• $CE = 10 \, \text{cm}$
• $EF = 11 \, \text{cm}$

Since $\triangle CEF$ is a right-angled triangle at $E$, we can use the Pythagorean theorem to find $CF$ (the hypotenuse $BC$):

$CF = BC = \sqrt{CE^2 + EF^2}$ Substitute the values: $BC = \sqrt{10^2 + 11^2} = \sqrt{100 + 121} = \sqrt{221}$ Calculating the square root and rounding to 1 decimal place: $BC \approx 14.9 \, \text{cm}$

Part (b): Finding the length of $BD$ using the exact answer for $BC$

Now that we know $BC = \sqrt{221} \, \text{cm}$ (exact form), we use triangle $\triangle BCD$, where:

• $BC = \sqrt{221} \, \text{cm}$
• $CD = 16 \, \text{cm}$

Again, we apply the Pythagorean theorem in $\triangle BCD$: $BD = \sqrt{BC^2 + CD^2} = \sqrt{(\sqrt{221})^2 + 16^2} = \sqrt{221 + 256} = \sqrt{477}$ Calculating the square root and rounding to 1 decimal place: $BD \approx 21.8 \, \text{cm}$

Final Answers

• (a) $BC \approx 14.9 \, \text{cm}$
• (b) $BD \approx 21.8 \, \text{cm}$

Would you like further details on any step?

Here are some related questions that might interest you:

1. How would the calculation change if one of the lengths was different?
2. What if we needed to find the angle $\angle BCF$ in $\triangle BCF$?
3. How can we verify these answers using trigonometry?
4. What would be the area of triangle $\triangle BCD$?
5. How would we find the volume of the entire triangular prism?

Tip: For any right-angled triangle, the Pythagorean theorem is your go-to tool for finding unknown side lengths if you know the other two.