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To solve this problem, we need to find:

1. The length of \( AC \).
2. The measure of \( \angle ADC \).

Let’s break it down step-by-step.

### Part (a): Finding the Length of \( AC \)

Given:
- \( AB = 6.2 \, \text{cm} \)
- \( BC = 5 \, \text{cm} \)
- \( \angle ABC = 50^\circ \)

Since we have a triangle \( ABC \) with two known sides and the included angle, we can use the **Cosine Rule** to find \( AC \). The Cosine Rule states:

\[
AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\angle ABC)
\]

Substitute the values:

\[
AC^2 = (6.2)^2 + (5)^2 - 2 \cdot 6.2 \cdot 5 \cdot \cos(50^\circ)
\]

Calculating each part:

1. \( (6.2)^2 = 38.44 \)
2. \( (5)^2 = 25 \)
3. \( 2 \cdot 6.2 \cdot 5 = 62 \)
4. \( \cos(50^\circ) \approx 0.6428 \)

Now plug in these values:

\[
AC^2 = 38.44 + 25 - 62 \cdot 0.6428
\]
\[
AC^2 = 63.44 - 39.8536
\]
\[
AC^2 \approx 23.5864
\]

Taking the square root:

\[
AC \approx \sqrt{23.5864} \approx 4.86 \, \text{cm}
\]

So, the length of \( AC \) is approximately \( 4.86 \, \text{cm} \).

### Part (b): Finding \( \angle ADC \)

Now, we have triangle \( ACD \) with:
- \( AC \approx 4.86 \, \text{cm} \)
- \( AD = 8 \, \text{cm} \)
- \( \angle CAD = 65^\circ \)

We can use the **Sine Rule** to find \( \angle ADC \). The Sine Rule states:

\[
\frac{AC}{\sin(\angle ADC)} = \frac{AD}{\sin(\angle CAD)}
\]

Rearrange to solve for \( \sin(\angle ADC) \):

\[
\sin(\angle ADC) = \frac{AC \cdot \sin(\angle CAD)}{AD}
\]

Substitute the values:

1. \( AC \approx 4.86 \)
2. \( AD = 8 \)
3. \( \sin(65^\circ) \approx 0.9063 \)

So,

\[
\sin(\angle ADC) = \frac{4.86 \cdot 0.9063}{8}
\]
\[
\sin(\angle ADC) \approx \frac{4.4067}{8}
\]
\[
\sin(\angle ADC) \approx 0.5508
\]

Now, find \( \angle ADC \) by taking the inverse sine:

\[
\angle ADC \approx \sin^{-1}(0.5508) \approx 33.4^\circ
\]

### Summary of Solutions

1. \( AC \approx 4.86 \, \text{cm} \)
2. \( \angle ADC \approx 33.4^\circ \)

Would you like further details on any of these steps, or have any questions?

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### Related Questions

1. How would the solution change if \( \angle ABC \) was different?
2. Can the Sine Rule be used if we only have two sides of a triangle?
3. What is the Law of Cosines, and when is it most useful?
4. How would you find \( \angle ACB \) in this triangle?
5. What would happen if \( AD \) was shorter than \( AC \)?

### Tip
When dealing with triangles, remember that the Cosine Rule is most useful for finding unknowns in non-right triangles when you know two sides and the included angle.

Find the length of AC and the angle ADC in the given quadrilateral.

Learn how to find the length of AC and angle ADC in a quadrilateral using trigonometric principles such as the Cosine Rule and Sine Rule. This problem includes calculations for lengths and angles, suitable for students in Grades 10-12.

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