To solve the problem of finding the distance between each pair of points using the Pythagorean Theorem, we will follow these steps:
Step-by-Step Solution:
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1. Identify the coordinates of the points:
- For each graph, identify the coordinates of the two points.
- Let the coordinates of the first point be \((x_1, y_1)\) and the coordinates of the second point be \((x_2, y_2)\).
####
2. Use the distance formula:
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
This formula is derived from the Pythagorean Theorem.
####
3. Calculate the distance for each pair of points:
---
Problem 1:
- Coordinates: \((-2, 4)\) and \((2, -2)\)
- Distance:
\[
d = \sqrt{(2 - (-2))^2 + (-2 - 4)^2} = \sqrt{(2 + 2)^2 + (-2 - 4)^2} = \sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]
---
Problem 2:
- Coordinates: \((-3, -2)\) and \((4, 3)\)
- Distance:
\[
d = \sqrt{(4 - (-3))^2 + (3 - (-2))^2} = \sqrt{(4 + 3)^2 + (3 + 2)^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}
\]
---
Problem 3:
- Coordinates: \((-1, 3)\) and \((3, -1)\)
- Distance:
\[
d = \sqrt{(3 - (-1))^2 + (-1 - 3)^2} = \sqrt{(3 + 1)^2 + (-1 - 3)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
---
Problem 4:
- Coordinates: \((-3, -2)\) and \((3, -2)\)
- Distance:
\[
d = \sqrt{(3 - (-3))^2 + (-2 - (-2))^2} = \sqrt{(3 + 3)^2 + (-2 + 2)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6
\]
---
Problem 5:
- Coordinates: \((-4, -2)\) and \((2, 2)\)
- Distance:
\[
d = \sqrt{(2 - (-4))^2 + (2 - (-2))^2} = \sqrt{(2 + 4)^2 + (2 + 2)^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
\]
---
Problem 6:
- Coordinates: \((3, 2)\) and \((3, -4)\)
- Distance:
\[
d = \sqrt{(3 - 3)^2 + (-4 - 2)^2} = \sqrt{0^2 + (-6)^2} = \sqrt{36} = 6
\]
---
Problem 7:
- Coordinates: \((1, 1)\) and \((4, 3)\)
- Distance:
\[
d = \sqrt{(4 - 1)^2 + (3 - 1)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]
---
Problem 8:
- Coordinates: \((1, 2)\) and \((1, -2)\)
- Distance:
\[
d = \sqrt{(1 - 1)^2 + (-2 - 2)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
\]
---
Final Answers:
\[
\boxed{
\begin{aligned}
1) & \quad 2\sqrt{13} \\
2) & \quad \sqrt{74} \\
3) & \quad 4\sqrt{2} \\
4) & \quad 6 \\
5) & \quad 2\sqrt{13} \\
6) & \quad 6 \\
7) & \quad \sqrt{13} \\
8) & \quad 4
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of distance between two points worksheet.
