Geometry worksheet covering midpoint, slope, and distance calculations for coordinate pairs.

Here's the complete solution for each set of points, using the formulas provided on the worksheet.

---

Problem 5: (7, 1) and (18, 59)

Let:
- \( (x_1, y_1) = (7, 1) \)
- \( (x_2, y_2) = (18, 59) \)

Distance:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(18 - 7)^2 + (59 - 1)^2} = \sqrt{11^2 + 58^2} = \sqrt{121 + 3364} = \sqrt{3485}
\]
✔ Distance: \( \sqrt{3485} \) (This does not simplify further since 3485 = 5 × 17 × 41 — no perfect square factors.)

Midpoint:
\[
\left( \frac{x_2 + x_1}{2}, \frac{y_2 + y_1}{2} \right) = \left( \frac{18 + 7}{2}, \frac{59 + 1}{2} \right) = \left( \frac{25}{2}, \frac{60}{2} \right) = \left( 12.5, 30 \right)
\]
✔ Midpoint: \( (12.5, 30) \) or \( \left( \frac{25}{2}, 30 \right) \)

Slope:
\[
\frac{y_2 - y_1}{x_2 - x_1} = \frac{59 - 1}{18 - 7} = \frac{58}{11}
\]
✔ Slope: \( \frac{58}{11} \)

---

Problem 6: (-2, 4) and (7, -3)

Let:
- \( (x_1, y_1) = (-2, 4) \)
- \( (x_2, y_2) = (7, -3) \)

Distance:
\[
\sqrt{(7 - (-2))^2 + (-3 - 4)^2} = \sqrt{(9)^2 + (-7)^2} = \sqrt{81 + 49} = \sqrt{130}
\]
✔ Distance: \( \sqrt{130} \) (Does not simplify — 130 = 2 × 5 × 13)

Midpoint:
\[
\left( \frac{-2 + 7}{2}, \frac{4 + (-3)}{2} \right) = \left( \frac{5}{2}, \frac{1}{2} \right) = (2.5, 0.5)
\]
✔ Midpoint: \( \left( \frac{5}{2}, \frac{1}{2} \right) \) or \( (2.5, 0.5) \)

Slope:
\[
\frac{-3 - 4}{7 - (-2)} = \frac{-7}{9}
\]
✔ Slope: \( -\frac{7}{9} \)

---

Problem 7: (3, 1) and (-1, 6)

Let:
- \( (x_1, y_1) = (3, 1) \)
- \( (x_2, y_2) = (-1, 6) \)

Distance:
\[
\sqrt{(-1 - 3)^2 + (6 - 1)^2} = \sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}
\]
✔ Distance: \( \sqrt{41} \) (Prime — doesn’t simplify)

Midpoint:
\[
\left( \frac{3 + (-1)}{2}, \frac{1 + 6}{2} \right) = \left( \frac{2}{2}, \frac{7}{2} \right) = (1, 3.5)
\]
✔ Midpoint: \( \left(1, \frac{7}{2}\right) \) or \( (1, 3.5) \)

Slope:
\[
\frac{6 - 1}{-1 - 3} = \frac{5}{-4} = -\frac{5}{4}
\]
✔ Slope: \( -\frac{5}{4} \)

---

Problem 8: (1, 4) and (6, 9)

Let:
- \( (x_1, y_1) = (1, 4) \)
- \( (x_2, y_2) = (6, 9) \)

Distance:
\[
\sqrt{(6 - 1)^2 + (9 - 4)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\]
✔ Distance: \( 5\sqrt{2} \)

Midpoint:
\[
\left( \frac{1 + 6}{2}, \frac{4 + 9}{2} \right) = \left( \frac{7}{2}, \frac{13}{2} \right) = (3.5, 6.5)
\]
✔ Midpoint: \( \left( \frac{7}{2}, \frac{13}{2} \right) \) or \( (3.5, 6.5) \)

Slope:
\[
\frac{9 - 4}{6 - 1} = \frac{5}{5} = 1
\]
✔ Slope: \( 1 \)

---

## ✔ Final Answers Summary:

5. (7,1) and (18,59)

- Distance: \( \sqrt{3485} \)
- Midpoint: \( (12.5, 30) \)
- Slope: \( \frac{58}{11} \)

6. (-2,4) and (7,-3)

- Distance: \( \sqrt{130} \)
- Midpoint: \( (2.5, 0.5) \)
- Slope: \( -\frac{7}{9} \)

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

- Distance: \( \sqrt{41} \)
- Midpoint: \( (1, 3.5) \)
- Slope: \( -\frac{5}{4} \)

8. (1,4) and (6,9)

- Distance: \( 5\sqrt{2} \)
- Midpoint: \( (3.5, 6.5) \)
- Slope: \( 1 \)

---

All work shown as requested. Let me know if you’d like these plotted or explained further!