https://file.aiquickdraw.com/mathbot/file/17267166957620q2aydq7.jpg

This diagram shows a right triangle, and our task is to find the value of \( x \), which is the base of the smaller triangle on the left. Let's break this down:

### Observations:
- The triangle is divided into two smaller right triangles by a perpendicular line.
- One triangle has angles of \( 60^\circ \), \( 90^\circ \), and \( 30^\circ \).
- The other triangle has angles of \( 45^\circ \), \( 45^\circ \), and \( 90^\circ \).
- The hypotenuse of the larger triangle is 15 units.

### Strategy:
1. **For the 30-60-90 triangle** (on the right):
   - The property of a 30-60-90 triangle is that the ratio of the sides opposite the angles \( 30^\circ \), \( 60^\circ \), and the hypotenuse is \( 1 : \sqrt{3} : 2 \). The hypotenuse is 15 units.
   - Let the side opposite the \( 30^\circ \) angle be \( a \), and the side opposite the \( 60^\circ \) angle be \( b \).

2. **For the 45-45-90 triangle** (on the left):
   - This triangle has both legs equal (because both angles are \( 45^\circ \)).
   - The ratio of the legs to the hypotenuse in a 45-45-90 triangle is \( 1:1:\sqrt{2} \). We can use this to calculate \( x \).

Let's first calculate the sides of the right triangle, then solve for \( x \). I'll compute these values now.Here are the results from the calculations:

1. The side opposite the \( 30^\circ \) angle is \( 7.5 \) units.
2. The side opposite the \( 60^\circ \) angle, which is shared by both triangles, is approximately \( 12.99 \) units.
3. The value of \( x \), the base of the 45-45-90 triangle, is approximately \( 9.19 \) units.

Would you like more details or have any other questions?

Here are five related questions to expand on this:

1. What is the total perimeter of the entire triangle?
2. Can you find the area of the larger triangle using the calculated values?
3. How would the calculations change if the hypotenuse was different, say 20 units?
4. How do the properties of special triangles (30-60-90 and 45-45-90) help simplify these calculations?
5. Can you calculate the height of the triangle from the base \( x \)?

**Tip:** Always remember the side length ratios of special triangles (30-60-90 and 45-45-90), as they simplify many geometric problems.

Find the value of x in the triangle, given the angles and side lengths.

Learn how to find the value of x in a triangle composed of a 30-60-90 and a 45-45-90 triangle. The problem involves applying trigonometric ratios and special triangle properties to solve for unknown sides.

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