Question

Which of the following statements correctly describes the points?
The distance between (3, 4) and (3, –7) is 11 units.
The distance between (3, 4) and (3, –7) is 3 units.
The distance between (4, 3) and (–7, 3) is 10 units.
The distance between (4, 3) and (4, –7) is 4 units.

Answer 1

To determine the distances between the given points, we can use the distance formula for two points in a plane \((x_1, y_1)\) and \((x_2, y_2)\):

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

However, we can simplify the process for vertical and horizontal distances:

1. **For the points (3, 4) and (3, –7)**:
These points have the same x-coordinate, meaning they are vertically aligned. The distance is simply the difference in the y-coordinates.
\[
d = |y_2 - y_1| = |–7 - 4| = |-11| = 11 \text{ units.}
\]

2. **For the points (4, 3) and (–7, 3)**:
These points have the same y-coordinate, meaning they are horizontally aligned. The distance is the difference in their x-coordinates.
\[
d = |x_2 - x_1| = |–7 - 4| = |–11| = 11 \text{ units.}
\]

3. **For the points (4, 3) and (4, –7)**:
These points are also vertically aligned.
\[
d = |y_2 - y_1| = |–7 - 3| = |-10| = 10 \text{ units.}
\]

Now, let's analyze the given statements:

1. **The distance between (3, 4) and (3, –7) is 11 units.** - **True**
2. **The distance between (3, 4) and (3, –7) is 3 units.** - **False**
3. **The distance between (4, 3) and (–7, 3) is 10 units.** - **False** (it is 11 units)
4. **The distance between (4, 3) and (4, –7) is 4 units.** - **False** (it is 10 units)

Thus, the only correct statement is:

**The distance between (3, 4) and (3, –7) is 11 units.**