To solve for $x$ in the triangle, we need to consider the geometry of the problem described. Here's the step-by-step process:

Given:

• A triangle with one side labeled $x$ (horizontal).
• The second side makes a 30° angle with the first side.
• The third side makes a 60° angle with the first side.
• There is a vertical line segment labeled 110 that connects a vertex to the first side at a right angle.

Assumptions based on the description:

1. The vertical segment labeled 110 is the height of the triangle (altitude) from the vertex opposite the base (horizontal side $x$) to the base itself.
2. The triangle is split by this height, making two right triangles.

To find $x$, we can use the properties of right triangles and trigonometry.

Strategy:

The height of 110 creates two right-angled triangles with angles of 30° and 60°.

In the triangle with a 30° angle:

• The height (110) is opposite the 30° angle.
• Use the tangent function, where:
  $\tan(30^\circ) = \frac{{\text{opposite}}}{{\text{adjacent}}} = \frac{110}{\text{adjacent side}}.$ From this, we can solve for the adjacent side (the distance from the point of the vertex to the line at a 30° angle).

For the triangle with a 60° angle:

• The height is also opposite the 60° angle.
• Use the tangent function: $\tan(60^\circ) = \frac{{\text{opposite}}}{{\text{adjacent}}} = \frac{110}{\text{adjacent side}}.$

By combining the results from these steps, we will be able to determine the value of $x$.

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