Find *x* in the figure below, rounded to one decimal place. A right triangle is given.

The first side is horizontal.

The second side of length 82 is at a right angle with the first side.

The third side is at an angle of 30° with the first side.

The triangle is split into 2 smaller triangles by a line segment connecting a vertex opposite the first side to a point on the first side at an angle 60°.

The part of the first side between the vertex of 30° and the point where the line segment ends is labeled *x*.

Given the description, we can approach the problem using trigonometric properties of right triangles. Here's the step-by-step breakdown of the process:

### Key details from the problem:
- We have a right triangle with one side of length 82 (which we'll call the vertical side).
- The third side (hypotenuse) forms a 30° angle with the first side (horizontal side).
- A line segment divides the triangle into two smaller triangles, and it forms a 60° angle with the first side.
- The length of the segment along the horizontal side between the 30° vertex and the point where the line ends is labeled \( x \).

We'll use trigonometric relationships to solve for \( x \).

### Step 1: Understanding the large triangle

We have a large right triangle where:
- One leg is the horizontal side (adjacent to the 30° angle).
- The vertical side is 82.
- The angle between the hypotenuse and the horizontal side is 30°.

Using trigonometric properties, we can find the length of the horizontal side. The tangent function relates the opposite side (vertical leg) to the adjacent side (horizontal leg).

\[
\tan(30^\circ) = \frac{\text{vertical side}}{\text{horizontal side}}
\]

\[
\tan(30^\circ) = \frac{82}{\text{horizontal side}}
\]

Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can substitute this in:

\[
\frac{1}{\sqrt{3}} = \frac{82}{\text{horizontal side}}
\]

Solving for the horizontal side:

\[
\text{horizontal side} = 82 \times \sqrt{3} \approx 82 \times 1.732 = 142.024
\]

Thus, the horizontal side of the large triangle is approximately 142.0 units.

### Step 2: Using the 60° smaller triangle

Now, the smaller triangle formed by the line segment has a 60° angle at the point where it meets the horizontal side. This smaller triangle shares the horizontal side with the larger triangle, and we want to find the portion of the horizontal side labeled \( x \).

The cosine of the 60° angle will relate the length of \( x \) to the length of the horizontal side.

\[
\cos(60^\circ) = \frac{x}{\text{horizontal side}}
\]

Since \( \cos(60^\circ) = \frac{1}{2} \), we substitute this in:

\[
\frac{1}{2} = \frac{x}{142.0}
\]

Solving for \( x \):

\[
x = 142.0 \times \frac{1}{2} = 71.0
\]

Thus, \( x \) is approximately \( 71.0 \) units.

### Final Answer:
The value of \( x \) is \( 71.0 \), rounded to one decimal place.

---

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do we calculate the horizontal side if the given angle changes from 30° to 45°?
2. What would be the length of the third side (hypotenuse) using the Pythagorean theorem?
3. How does the tangent function help in finding the length of a side in a right triangle?
4. What is the significance of using the cosine function for the smaller triangle?
5. How would the answer change if the vertical side were shorter, say 60 units?

**Tip:** Always double-check the units and make sure the angle you are working with is applied correctly to avoid errors in trigonometric problems.

Find x in the figure below, rounded to one decimal place. A right triangle is given. The first side is horizontal. The second side of length 82 is at a right angle with the first side. The third side is at an angle of 30° with the first side. The triangle is split into 2 smaller triangles by a line segment connecting a vertex opposite the first side to a point on the first side at an angle of 60°. The part of the first side between the vertex of 30° and the point where the line segment ends is labeled x.

Learn how to solve for x in a right triangle where one side is 82 units long and the angle between the hypotenuse and the horizontal side is 30°. Using trigonometric identities such as tangent and cosine, this solution breaks down the steps to find the segment of the horizontal side labeled x. Suitable for high school students.

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