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To calculate the size of angle \( \angle BCD \), we can use the cosine rule, as we are given three sides of triangle \( BCD \):

- \( BC = 7.4 \) cm
- \( BD = 6.5 \) cm
- \( CD = 4.7 \) cm

The cosine rule is given by:
\[
\cos(C) = \frac{b^2 + c^2 - a^2}{2bc}
\]
where:
- \( C \) is the angle we are trying to find (\( \angle BCD \)),
- \( a \) is the side opposite the angle (\( BD \)),
- \( b \) and \( c \) are the other two sides (\( BC \) and \( CD \)).

Substitute the known values into the formula:
\[
\cos(C) = \frac{6.5^2 + 4.7^2 - 7.4^2}{2 \times 6.5 \times 4.7}
\]

First, calculate each term:
- \( 6.5^2 = 42.25 \)
- \( 4.7^2 = 22.09 \)
- \( 7.4^2 = 54.76 \)

Now substitute:
\[
\cos(C) = \frac{42.25 + 22.09 - 54.76}{2 \times 6.5 \times 4.7}
\]
\[
\cos(C) = \frac{9.58}{61.1}
\]
\[
\cos(C) \approx 0.1568
\]

To find the angle \( C \):
\[
C = \cos^{-1}(0.1568) \approx 80.98^\circ
\]

Rounding to 1 decimal place, the size of angle \( BCD \) is approximately:
\[
\boxed{81.0^\circ}
\]

Let me know if you would like further details or clarifications!

Here are 5 related questions:
1. How would you calculate the angle \( \angle BDA \) using the cosine rule?
2. Can the sine rule be applied to this triangle? If so, how would you use it?
3. How would the answer change if side \( BC \) were shorter?
4. What are the other angles in the quadrilateral \( ABCD \)?
5. How can you use the Pythagorean theorem to verify the distances?

**Tip:** Always double-check your units and rounding when working with angles and lengths in trigonometry.

Calculate the size of angle BCD in the quadrilateral ABCD where AB = 4.7 cm, BD = 6.5 cm, and CD = 7.4 cm.

Learn how to calculate the size of angle BCD in a quadrilateral using the cosine rule. Given sides AB = 4.7 cm, BD = 6.5 cm, and CD = 7.4 cm, this problem involves the application of trigonometry concepts and cosine rule. Suitable for students in Grade 10-12.

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