Let's break down the problem and solve it step by step.

Given:

• $\triangle ABC$ and $\triangle PCD$ are right-angled triangles.
• Angle $\angle ABC = 40^\circ$
• $AB = 10$ cm, $PD = 8$ cm, $BD = 15$ cm.

We need to calculate:

1. The length of $BC$ in centimeters.
2. The size of angle $PDC$ in degrees.

i) Finding the length of $BC$:

Using right-angled triangle $ABC$, we can apply trigonometric ratios. Since we know the angle $\angle ABC = 40^\circ$, and we are given the adjacent side $AB = 10 \, \text{cm}$, we will use the tangent ratio:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Here:

• Opposite side is $BC$,
• Adjacent side is $AB = 10 \, \text{cm}$,
• $\theta = 40^\circ$.

$\tan(40^\circ) = \frac{BC}{10}$

Thus, solving for $BC$:

$BC = 10 \times \tan(40^\circ)$

Using a calculator:

$BC \approx 10 \times 0.8391 \approx 8.39 \, \text{cm}$

So, the length of $BC$ is approximately 8.4 cm (to 1 decimal place).

ii) Finding the size of angle $PDC$:

We now look at triangle $PCD$. We know that:

• $PD = 8 \, \text{cm}$,
• $BD = 15 \, \text{cm}$, so $CD = BD - BC = 15 - 8.39 \approx 6.61 \, \text{cm}.$

Now, we can use the tangent ratio in triangle $PCD$ to find $\angle PDC$:

$\tan(\angle PDC) = \frac{PC}{CD}$

Since $PC = PD = 8 \, \text{cm}$ and $CD \approx 6.61 \, \text{cm}$, we have:

$\tan(\angle PDC) = \frac{8}{6.61} \approx 1.210$

Taking the inverse tangent (arctan) to find the angle:

$\angle PDC = \tan^{-1}(1.210) \approx 50.2^\circ$

Thus, the size of angle $PDC$ is approximately 50.2 degrees.

Summary:

1. The length of $BC$ is approximately 8.4 cm.
2. The size of angle $PDC$ is approximately 50.2 degrees.

Would you like a more detailed explanation for any part of the solution?
Here are five questions to further explore this topic:

1. How would the calculations change if the angle $\angle ABC$ was different?
2. Can you explain how the tangent ratio is used to solve for sides and angles?
3. What is the difference between the sine, cosine, and tangent ratios in trigonometry?
4. How would you solve for unknown sides using the Pythagorean theorem?
5. How do inverse trigonometric functions help in finding angles?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) based on the angle unit you're working with.