Let's solve each problem step by step using the
Distance Formula and the
Midpoint Formula.
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🔷 Distance Formula:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
🔷 Midpoint Formula:
$$
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$
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##
✔ Part 1: Find the distance between each set of points
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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 units
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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: $\sqrt{101}$ units ≈ 10.05 units
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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: $2\sqrt{26}$ units ≈ 10.2 units
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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: $5\sqrt{10}$ units ≈ 15.81 units
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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: $6\sqrt{2}$ units ≈ 8.49 units
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##
✔ Part 2: Find the midpoint for each line segment
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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)
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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)
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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)
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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)
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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)
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##
✔ 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)
7. (2.5, 2)
8. (-1, 4)
9. (0.5, 0.5)
10. (1, -1)
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Let me know if you'd like this formatted as a printable worksheet or need explanations in simpler terms!
Parent Tip: Review the logic above to help your child master the concept of distance formula worksheet 1.
