Let's solve each problem step by step using the
Distance Formula and the
Midpoint Formula.
---
🔷 Distance Formula:
For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ is:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
---
🔷 Midpoint Formula:
The midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:
$$
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$
---
##
✔ Part 1: Find the Distance Between Each Set of Points
---
1. (0, 0) and (4, 3)
$$
d = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
$$
✔ Answer: 5
---
2. (3, -3) and (2, 7)
$$
d = \sqrt{(2 - 3)^2 + (7 - (-3))^2} = \sqrt{(-1)^2 + (10)^2} = \sqrt{1 + 100} = \sqrt{101}
$$
✔ Answer: $\sqrt{101}$ (≈ 10.05)
---
3. (4, 5) and (-6, 3)
$$
d = \sqrt{(-6 - 4)^2 + (3 - 5)^2} = \sqrt{(-10)^2 + (-2)^2} = \sqrt{100 + 4} = \sqrt{104} = 2\sqrt{26}
$$
✔ Answer: $2\sqrt{26}$ (≈ 10.2)
---
4. (-2, 8) and (3, -7)
$$
d = \sqrt{(3 - (-2))^2 + (-7 - 8)^2} = \sqrt{(5)^2 + (-15)^2} = \sqrt{25 + 225} = \sqrt{250} = 5\sqrt{10}
$$
✔ Answer: $5\sqrt{10}$ (≈ 15.81)
---
5. (4, 2) and (-2, -4)
$$
d = \sqrt{(-2 - 4)^2 + (-4 - 2)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
$$
✔ Answer: $6\sqrt{2}$ (≈ 8.49)
---
##
✔ Part 2: Find the Midpoint for Each Line Segment
---
6. (0, 0) and (4, 3)
$$
M = \left( \frac{0 + 4}{2}, \frac{0 + 3}{2} \right) = \left( \frac{4}{2}, \frac{3}{2} \right) = (2, 1.5)
$$
✔ Answer: (2, 1.5) or $(2, \frac{3}{2})$
---
7. (3, -3) and (2, 7)
$$
M = \left( \frac{3 + 2}{2}, \frac{-3 + 7}{2} \right) = \left( \frac{5}{2}, \frac{4}{2} \right) = (2.5, 2)
$$
✔ Answer: (2.5, 2) or $\left(\frac{5}{2}, 2\right)$
---
8. (4, 5) and (-6, 3)
$$
M = \left( \frac{4 + (-6)}{2}, \frac{5 + 3}{2} \right) = \left( \frac{-2}{2}, \frac{8}{2} \right) = (-1, 4)
$$
✔ Answer: (-1, 4)
---
9. (-2, 8) and (3, -7)
$$
M = \left( \frac{-2 + 3}{2}, \frac{8 + (-7)}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right) = (0.5, 0.5)
$$
✔ Answer: (0.5, 0.5) or $\left(\frac{1}{2}, \frac{1}{2}\right)$
---
10. (4, 2) and (-2, -4)
$$
M = \left( \frac{4 + (-2)}{2}, \frac{2 + (-4)}{2} \right) = \left( \frac{2}{2}, \frac{-2}{2} \right) = (1, -1)
$$
✔ Answer: (1, -1)
---
##
✔ Final Answers Summary:
Distances:
1. 5
2. $\sqrt{101}$
3. $2\sqrt{26}$
4. $5\sqrt{10}$
5. $6\sqrt{2}$
Midpoints:
6. $(2, 1.5)$ or $(2, \frac{3}{2})$
7. $(2.5, 2)$ or $(\frac{5}{2}, 2)$
8. $(-1, 4)$
9. $(0.5, 0.5)$ or $(\frac{1}{2}, \frac{1}{2})$
10. $(1, -1)$
---
Let me know if you'd like this formatted as a printable PDF or need it explained in another way!
Parent Tip: Review the logic above to help your child master the concept of distance formula practice worksheet.
