Question

The vertices of a rectangle are located at A(4, -1), B(-4, -1), C(-4, 6), and D(4, 6). What is the distance between the side AB and BC respectively?

Answer 1

To determine the distances between the sides AB and BC of the rectangle, we can analyze their geometrical properties based on their coordinates.

1. **Identify the coordinates of the vertices:**
- A(4, -1)
- B(-4, -1)
- C(-4, 6)
- D(4, 6)

2. **Determine the lengths of the sides:**
- Side AB runs between points A(4, -1) and B(-4, -1). The y-coordinates are the same, indicating that AB is horizontal.
- Side BC runs between points B(-4, -1) and C(-4, 6). The x-coordinates are the same, indicating that BC is vertical.

3. **Calculate the lengths of sides AB and BC:**
- The length of side AB can be calculated using the formula for the distance between two points:
\[
\text{Length of AB} = |x_2 - x_1| = |-4 - 4| = 8
\]
- The length of side BC is determined by the difference in the y-coordinates:
\[
\text{Length of BC} = |y_2 - y_1| = |6 - (-1)| = 7
\]

4. **Finding the distances:**
- **Distance between side AB and BC:** Since AB is horizontal and BC is vertical, the distance is essentially the direction along the x-axis from the line y = -1 (where AB is) to the line x = -4 (where BC is). We recognize that both line segments intersect at point B.
- Since they meet and are perpendicular, the distance between them is simply the horizontal distance from point B to the vertical line going through point A's x-coordinate. The distance from B (-4, -1) to line x = 4 is:
\[
\text{Distance} = |4 - (-4)| = 8
\]

In conclusion, the lengths of the sides are as follows:
- Distance between side AB (horizontal) and side BC (vertical): 8 (along the horizontal).
- Distance between side BC (vertical) and side AB (horizontal): 7 (along the vertical).

Thus, the relation can be stated as follows:
- Distance from side AB to side BC = 8 units.
- Distance from side BC to side AB = 7 units.