Find the distance between the points. Give an exact answer and an approximation to three decimal places.
​(1,-4) and ​(-3,-2)

To find the distance between the points \( (1, -4) \) and \( (-3, -2) \), we use the distance formula:

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

Where:
- \( (x_1, y_1) = (1, -4) \)
- \( (x_2, y_2) = (-3, -2) \)

Now, substitute the coordinates into the formula:

\[
d = \sqrt{(-3 - 1)^2 + (-2 + 4)^2}
\]
\[
d = \sqrt{(-4)^2 + (2)^2}
\]
\[
d = \sqrt{16 + 4}
\]
\[
d = \sqrt{20}
\]

The exact distance is \( \sqrt{20} \). Simplifying \( \sqrt{20} \):

\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}
\]

So, the exact distance is \( 2\sqrt{5} \).

Next, let's approximate this value to three decimal places:

\[
2\sqrt{5} \approx 2 \times 2.236 = 4.472
\]

Therefore, the distance between the points is:
- Exact: \( 2\sqrt{5} \)
- Approximate: \( 4.472 \)

Would you like more details on this calculation, or do you have any other questions?

### Related Questions:
1. What is the distance between points \( (2, -3) \) and \( (-1, 4) \)?
2. How do you find the midpoint of two points?
3. What is the distance between \( (0, 0) \) and \( (3, 4) \)?
4. Can you derive the distance formula from the Pythagorean theorem?
5. How do you find the distance between points in three dimensions?

### Tip:
Always check if the square root can be simplified further to get the most exact form before approximating.

Learn how to find the distance between points (1, -4) and (-3, -2) using the distance formula. Understand the application of the Pythagorean theorem in coordinate geometry. Suitable for students in Grades 7-9.

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