The side BC of an equilateral `angleABC` is parallel to the x-axis. What are the slope of its sides?

Answer

To find the slopes of the sides of the equilateral triangle ABC, where side BC is parallel to the x-axis, we can follow these steps:

### Step 1: Determine the slope of side BC
Since side BC is parallel to the x-axis, its slope can be calculated as follows:
- The angle of inclination of BC with the x-axis is 0 degrees.
- The slope (m) is given by the formula:
\[ m = \tan(\theta) \]
where \(\theta\) is the angle of inclination.

Calculating the slope:
\[ m_{BC} = \tan(0^\circ) = 0 \]

### Step 2: Determine the slope of side AC
For side AC, we need to consider the angle it makes with the positive direction of the x-axis. In an equilateral triangle, each angle is 60 degrees. Since AC is inclined upwards from point A to point C, we can take the angle of inclination as 60 degrees.

Calculating the slope:
\[ m_{AC} = \tan(60^\circ) = \sqrt{3} \]

### Step 3: Determine the slope of side AB
For side AB, we need to consider the angle it makes with the positive direction of the x-axis. The angle at point B is 60 degrees, but since we are measuring in the positive direction, we need to consider the exterior angle, which is 120 degrees.

Calculating the slope:
\[ m_{AB} = \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3} \]

### Summary of Slopes
- Slope of BC: \( m_{BC} = 0 \)
- Slope of AC: \( m_{AC} = \sqrt{3} \)
- Slope of AB: \( m_{AB} = -\sqrt{3} \)

### Final Answer
The slopes of the sides of triangle ABC are:
- Slope of BC: \( 0 \)
- Slope of AC: \( \sqrt{3} \)
- Slope of AB: \( -\sqrt{3} \)