https://file.aiquickdraw.com/mathbot/file/1730064563866hjy4cdl6.jpg

To solve each question, we'll use the distance formula:

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

Let's go through each problem on the worksheet.

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### Problem Solutions

1. **Points (4, -6) and (-5, -6)**  
   \[
   \text{Distance} = \sqrt{(-5 - 4)^2 + (-6 - (-6))^2} = \sqrt{(-9)^2 + 0^2} = \sqrt{81} = 9 \text{ units}
   \]

2. **Points (6, -7) and (-5, -4)**  
   \[
   \text{Distance} = \sqrt{(-5 - 6)^2 + (-4 - (-7))^2} = \sqrt{(-11)^2 + 3^2} = \sqrt{121 + 9} = \sqrt{130} \approx 11.4 \text{ units}
   \]

3. **Points (-12, 3) and (-4, -4)**  
   \[
   \text{Distance} = \sqrt{(-4 - (-12))^2 + (-4 - 3)^2} = \sqrt{8^2 + (-7)^2} = \sqrt{64 + 49} = \sqrt{113} \approx 10.6 \text{ units}
   \]

4. **Points (-7, -4) and (-7, 9)**  
   \[
   \text{Distance} = \sqrt{(-7 - (-7))^2 + (9 - (-4))^2} = \sqrt{0^2 + 13^2} = \sqrt{169} = 13 \text{ units}
   \]

5. **Points (24, 18) and (-6, 18)**  
   \[
   \text{Distance} = \sqrt{(-6 - 24)^2 + (18 - 18)^2} = \sqrt{(-30)^2 + 0^2} = \sqrt{900} = 30 \text{ units}
   \]

6. **Points (12, -6) and (12, -21)**  
   \[
   \text{Distance} = \sqrt{(12 - 12)^2 + (-21 - (-6))^2} = \sqrt{0^2 + (-15)^2} = \sqrt{225} = 15 \text{ units}
   \]

7. **Points (0, -6) and (10, -6)**  
   \[
   \text{Distance} = \sqrt{(10 - 0)^2 + (-6 - (-6))^2} = \sqrt{10^2 + 0^2} = \sqrt{100} = 10 \text{ units}
   \]

8. **Points (-3, 5) and (-3, 77)**  
   \[
   \text{Distance} = \sqrt{(-3 - (-3))^2 + (77 - 5)^2} = \sqrt{0^2 + 72^2} = \sqrt{5184} = 72 \text{ units}
   \]

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Would you like further details on any of these solutions or additional assistance?

### Related Questions

1. What is the general formula for finding the distance between two points in three-dimensional space?
2. How do we apply the distance formula to points on a coordinate plane?
3. What are some applications of the distance formula in real-world contexts?
4. How can the distance formula be adapted to work with complex numbers?
5. How do changes in the x-coordinates or y-coordinates of points affect the calculated distance?

### Tip
Always double-check your coordinates before plugging them into the distance formula. Misplacing even one sign can lead to an incorrect answer.

Find the distance between each pair of points.

Solve how to calculate the distance between pairs of points using the distance formula. Detailed steps show distance calculation for points such as (4, -6) and (-5, -6) or (-3, 5) and (-3, 77). Ideal for students in Grades 7-9 studying coordinate geometry.

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