For #9-14 find the distance, leave your answers in simplified radical form.
9. ( 0,3) ( -4, 6) 10. (-7, 3) ( -6, 7) 11. (-3,-6) (-2,1)
12. (4, 7) (-2, -5) 13. (7,-6) ( -4, -7) 14.   2 3,5 3, 2
15. What is the formula for midpoint?___________________________
For # 16-21 find the midpoint.
16. (-10, -5) ( 4,-1) 17. ( -4, -9) (-10,-11) 18. (3, -5) (0,-2)
19. (9, -1) (10, -11) 20. (-9,-4) (3, -3) 21. ( -12, -3) (7, 9)


Let's go through each problem step-by-step.

### Problems 9-14: Finding the Distance

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:

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

Let's calculate each one:

#### 9. Points: \((0, 3)\) and \((-4, 6)\)

\[
\text{Distance} = \sqrt{(-4 - 0)^2 + (6 - 3)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

#### 10. Points: \((-7, 3)\) and \((-6, 7)\)

\[
\text{Distance} = \sqrt{(-6 - (-7))^2 + (7 - 3)^2} = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}
\]

#### 11. Points: \((-3, -6)\) and \((-2, 1)\)

\[
\text{Distance} = \sqrt{(-2 - (-3))^2 + (1 - (-6))^2} = \sqrt{(1)^2 + (7)^2} = \sqrt{1 + 49} = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
\]

#### 12. Points: \((4, 7)\) and \((-2, -5)\)

\[
\text{Distance} = \sqrt{(-2 - 4)^2 + (-5 - 7)^2} = \sqrt{(-6)^2 + (-12)^2} = \sqrt{36 + 144} = \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
\]

#### 13. Points: \((7, -6)\) and \((-4, -7)\)

\[
\text{Distance} = \sqrt{(-4 - 7)^2 + (-7 - (-6))^2} = \sqrt{(-11)^2 + (-1)^2} = \sqrt{121 + 1} = \sqrt{122} = \sqrt{2 \times 61} = \sqrt{122}
\]

#### 14. Points: \((2\sqrt{3}, 5)\) and \((3\sqrt{3}, 2)\)

\[
\text{Distance} = \sqrt{(3\sqrt{3} - 2\sqrt{3})^2 + (2 - 5)^2} = \sqrt{(\sqrt{3})^2 + (-3)^2} = \sqrt{3 + 9} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]

### Problem 15: Formula for Midpoint

The formula for the midpoint \((x_m, y_m)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

### Problems 16-21: Finding the Midpoint

Using the midpoint formula:

#### 16. Points: \((-10, -5)\) and \((4, -1)\)

\[
\text{Midpoint} = \left(\frac{-10 + 4}{2}, \frac{-5 + (-1)}{2}\right) = \left(\frac{-6}{2}, \frac{-6}{2}\right) = (-3, -3)
\]

#### 17. Points: \((-4, -9)\) and \((-10, -11)\)

\[
\text{Midpoint} = \left(\frac{-4 + (-10)}{2}, \frac{-9 + (-11)}{2}\right) = \left(\frac{-14}{2}, \frac{-20}{2}\right) = (-7, -10)
\]

#### 18. Points: \((3, -5)\) and \((0, -2)\)

\[
\text{Midpoint} = \left(\frac{3 + 0}{2}, \frac{-5 + (-2)}{2}\right) = \left(\frac{3}{2}, \frac{-7}{2}\right) = \left(\frac{3}{2}, -\frac{7}{2}\right)
\]

#### 19. Points: \((9, -1)\) and \((10, -11)\)

\[
\text{Midpoint} = \left(\frac{9 + 10}{2}, \frac{-1 + (-11)}{2}\right) = \left(\frac{19}{2}, \frac{-12}{2}\right) = \left(\frac{19}{2}, -6\right)
\]

#### 20. Points: \((-9, -4)\) and \((3, -3)\)

\[
\text{Midpoint} = \left(\frac{-9 + 3}{2}, \frac{-4 + (-3)}{2}\right) = \left(\frac{-6}{2}, \frac{-7}{2}\right) = \left(-3, -\frac{7}{2}\right)
\]

#### 21. Points: \((-12, -3)\) and \((7, 9)\)

\[
\text{Midpoint} = \left(\frac{-12 + 7}{2}, \frac{-3 + 9}{2}\right) = \left(\frac{-5}{2}, \frac{6}{2}\right) = \left(-\frac{5}{2}, 3\right)
\]

Would you like further details on any of these problems or any additional explanations?

---

Here are five related questions to practice:

1. Find the distance between the points \((2, 4)\) and \((8, 7)\).
2. What is the distance between \((3, -2)\) and \((-5, -6)\)?
3. Determine the midpoint between the points \((7, 11)\) and \((2, -1)\).
4. Calculate the distance between \((4, 3)\) and \((-4, -3)\).
5. Find the midpoint of \((5, 9)\) and \((-7, 1)\).

**Tip:** Remember, for both distance and midpoint problems, carefully substitute the coordinates into the respective formulas, ensuring signs are handled correctly!

Learn how to solve problems on distance and midpoint in coordinate geometry. This page provides step-by-step solutions to finding distances between points and calculating midpoints using the distance formula and midpoint formula. Suitable for high school students.

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