https://file.aiquickdraw.com/mathbot/file/1727463558283i7dupz2o.jpg

It seems that the problem involves a triangle where we are tasked to find the length of side \( CD \). Here's a breakdown based on the image:

- \( AB = 6 \, \text{cm} \)
- \( AC = 12 \, \text{cm} \)
- \( \angle DAB = 55^\circ \)
- We need to calculate the length of \( CD \) to 3 significant figures.

### Steps to Solve:
1. **Use the Law of Cosines:**
   Since we have the angle between two known sides \( AB \) and \( AC \), we can apply the law of cosines:
   \[
   CD^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle DAB)
   \]
2. **Substitute the values:**
   \[
   CD^2 = 6^2 + 12^2 - 2 \cdot 6 \cdot 12 \cdot \cos(55^\circ)
   \]
3. **Simplify the equation** and solve for \( CD \).

Let me calculate this for you.The length of \( CD \) is approximately \( 9.87 \, \text{cm} \) when rounded to 3 significant figures.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the law of cosines relate to the Pythagorean theorem?
2. What happens when the angle in the law of cosines is \( 90^\circ \)?
3. Can you apply the law of sines to this triangle? If so, how?
4. How can we solve a triangle when we know only two angles and one side?
5. What is the significance of using radians in trigonometric calculations?

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

AB = 6 cm, AC = 12 cm, ∠DAB = 55°
Calculate the length of CD. Give your answer to 3 significant figures.

Learn how to find the length of side CD in a triangle using the Law of Cosines. Given AB = 6 cm, AC = 12 cm, and ∠DAB = 55°, this problem explores key trigonometric concepts and the law of cosines theorem. Suitable for students in Grades 9-10.

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