https://file.aiquickdraw.com/mathbot/file/1727187752850ix96bvyf.jpg

The image shows a triangle divided into two right-angled triangles, with the following details:

- One of the right triangles has an angle of 50° and a hypotenuse of 18.
- The other right triangle has an angle of 20°.
- The base of the large triangle is labeled \( x \), which is the length we need to find.

### Plan to solve for \( x \):

We can break the problem into two parts:

1. **Using trigonometry on the left right triangle:**
   - The left triangle has a hypotenuse of 18 and an angle of 50°. Using the cosine rule, we can find the adjacent side, which is half of \( x \).

2. **Using trigonometry on the right right triangle:**
   - The right triangle has an angle of 20°. We can use the tangent function to relate the height to the half of the base.

Let's solve for \( x \).The total length of the base \( x \) is approximately 49.45 units.

Would you like further clarification or details on the steps? Here are five related questions you might explore:

1. How can the sine or cosine rules be applied in non-right triangles?
2. What are the key properties of right triangles in trigonometry?
3. How does the tangent function relate to the angle and sides of a triangle?
4. What other methods can be used to solve for unknown sides in a triangle?
5. How does changing the angle affect the side lengths in a right triangle?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems!

Find the length of the base (x) in a triangle with angles 50° and 20° and one side (hypotenuse) measuring 18 units, where the triangle is divided into two right-angled triangles.

Solve for the base (x) of a triangle with angles 50° and 20° using trigonometry. This problem involves applying the cosine and tangent functions and is suitable for high school students in Grades 10-12.

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