Completed geometry homework on distance and midpoint formulas.

Let's go through the homework step by step and verify or correct each answer using the Distance Formula and 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 pair of points

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1. (-4, 6) and (3, -7)

$$
d = \sqrt{(3 - (-4))^2 + (-7 - 6)^2} = \sqrt{(7)^2 + (-13)^2} = \sqrt{49 + 169} = \sqrt{218}
$$

✔ Answer: $\boxed{\sqrt{218}}$
✘ Your answer: $\sqrt{716}$ → Incorrect

> Note: $7^2 = 49$, $(-13)^2 = 169$, $49+169=218$

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2. (-6, -5) and (2, 0)

$$
d = \sqrt{(2 - (-6))^2 + (0 - (-5))^2} = \sqrt{(8)^2 + (5)^2} = \sqrt{64 + 25} = \sqrt{89}
$$

✔ Answer: $\boxed{\sqrt{89}}$
✘ Your answer: $\sqrt{84}$ → Incorrect

> $8^2 = 64$, $5^2 = 25$, $64+25=89$

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3. (-1, 4) and (1, -1)

$$
d = \sqrt{(1 - (-1))^2 + (-1 - 4)^2} = \sqrt{(2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}
$$

✔ Answer: $\boxed{\sqrt{29}}$
✔ Your answer: $\sqrt{29}$ → Correct!

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4. (0, -8) and (3, 2)

$$
d = \sqrt{(3 - 0)^2 + (2 - (-8))^2} = \sqrt{(3)^2 + (10)^2} = \sqrt{9 + 100} = \sqrt{109}
$$

✔ Answer: $\boxed{\sqrt{109}}$
✔ Your answer: $\sqrt{109}$ → Correct!

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5. Graphed points (from grid)

Looking at the graph:
- One point is at $(-4, -4)$
- The other is at $(2, 2)$

Now calculate:

$$
d = \sqrt{(2 - (-4))^2 + (2 - (-4))^2} = \sqrt{(6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72}
$$

✔ Answer: $\boxed{\sqrt{72}}$
✔ Your answer: $\sqrt{72}$ → Correct!

> You can simplify $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$, but $\sqrt{72}$ is acceptable unless simplified.

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## ✔ Part 2: Find the midpoint of the segment given two endpoints

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6. A(2, 8) and B(-1, -4)

$$
M = \left( \frac{2 + (-1)}{2}, \frac{8 + (-4)}{2} \right) = \left( \frac{1}{2}, \frac{4}{2} \right) = \left( 0.5, 2 \right)
$$

✔ Answer: $\boxed{\left( \frac{1}{2}, 2 \right)}$
✘ Your answer: $(2, 2)$ → Incorrect

> Mistake: Average of $x$: $(2 + (-1))/2 = 1/2$, not 2.

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7. M(5, 9) and N(3, 7)

$$
M = \left( \frac{5 + 3}{2}, \frac{9 + 7}{2} \right) = \left( \frac{8}{2}, \frac{16}{2} \right) = (4, 8)
$$

✔ Answer: $\boxed{(4, 8)}$
✘ Your answer: $(6, 4)$ → Incorrect

> You might have added instead of averaged? $ (5+3)/2 = 4 $, not 6.

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8. P(2, 7) and Q(3, -5)

$$
M = \left( \frac{2 + 3}{2}, \frac{7 + (-5)}{2} \right) = \left( \frac{5}{2}, \frac{2}{2} \right) = \left( 2.5, 1 \right)
$$

✔ Answer: $\boxed{\left( \frac{5}{2}, 1 \right)}$
✘ Your answer: $(0, -6)$ → Incorrect

> This is way off — likely a calculation error.

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9. P(2, -6) and Q(4, 8)

$$
M = \left( \frac{2 + 4}{2}, \frac{-6 + 8}{2} \right) = \left( \frac{6}{2}, \frac{2}{2} \right) = (3, 1)
$$

✔ Answer: $\boxed{(3, 1)}$
✘ Your answer: $(-3, 1)$ → Incorrect

> Sign error on x-coordinate: $ (2+4)/2 = 3 $, not $-3$

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## ✔ Final Summary of Corrections

| Problem | Your Answer | Correct Answer |
|--------|-------------|----------------|
| 1 | $\sqrt{716}$ | $\boxed{\sqrt{218}}$ |
| 2 | $\sqrt{84}$ | $\boxed{\sqrt{89}}$ |
| 3 | $\sqrt{29}$ | ✔ $\boxed{\sqrt{29}}$ |
| 4 | $\sqrt{109}$ | ✔ $\boxed{\sqrt{109}}$ |
| 5 | $\sqrt{72}$ | ✔ $\boxed{\sqrt{72}}$ |
| 6 | $(2, 2)$ | $\boxed{\left(\frac{1}{2}, 2\right)}$ |
| 7 | $(6, 4)$ | $\boxed{(4, 8)}$ |
| 8 | $(0, -6)$ | $\boxed{\left(\frac{5}{2}, 1\right)}$ |
| 9 | $(-3, 1)$ | $\boxed{(3, 1)}$ |

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## 📌 Tips to Avoid Errors:

- Double-check signs when subtracting coordinates.
- For midpoints, always average both coordinates.
- Remember: $ (x_1 + x_2)/2 $, not just adding.
- Use parentheses: e.g., $(-1 - 4) = -5$, not $-1 - 4 = -5$ (correct), but don’t forget negative signs.

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Would you like me to provide a corrected version of the worksheet with all answers filled in neatly?