Q1P4 Midpoint and Distance Formulas - BulldogMath.com - Free Printable

Let's solve the problems step by step using the Midpoint Formula and Distance Formula, as requested.

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🔷 Midpoint Formula:

For two points $ (x_1, y_1) $ and $ (x_2, y_2) $, the midpoint $ M $ is:

$$
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

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🔷 Distance Formula:

The distance between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

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We'll go through each problem.

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## ✔ Part 1: Find the midpoint of the line segment with the given endpoints.

1) $(-5, -5), (10, -3)$

$$
\left( \frac{-5 + 10}{2}, \frac{-5 + (-3)}{2} \right) = \left( \frac{5}{2}, \frac{-8}{2} \right) = \left( 2.5, -4 \right)
$$
✔ Answer: $ (2.5, -4) $

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2) $ (1, 9), (1, 5) $

$$
\left( \frac{1+1}{2}, \frac{9+5}{2} \right) = (1, 7)
$$
✔ Answer: $ (1, 7) $

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3) $ (8, -6), (1, 4) $

$$
\left( \frac{8+1}{2}, \frac{-6+4}{2} \right) = \left( \frac{9}{2}, \frac{-2}{2} \right) = (4.5, -1)
$$
✔ Answer: $ (4.5, -1) $

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4) $ (-2, -9), (-5, 8) $

$$
\left( \frac{-2 + (-5)}{2}, \frac{-9 + 8}{2} \right) = \left( \frac{-7}{2}, \frac{-1}{2} \right) = (-3.5, -0.5)
$$
✔ Answer: $ (-3.5, -0.5) $

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5) $ (-9, 7), (-4, 8) $

$$
\left( \frac{-9 + (-4)}{2}, \frac{7 + 8}{2} \right) = \left( \frac{-13}{2}, \frac{15}{2} \right) = (-6.5, 7.5)
$$
✔ Answer: $ (-6.5, 7.5) $

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6) $ (-10, -10), (-6, -6) $

$$
\left( \frac{-10 + (-6)}{2}, \frac{-10 + (-6)}{2} \right) = \left( \frac{-16}{2}, \frac{-16}{2} \right) = (-8, -8)
$$
✔ Answer: $ (-8, -8) $

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7) $ (4, -3), (-5, -6) $

$$
\left( \frac{4 + (-5)}{2}, \frac{-3 + (-6)}{2} \right) = \left( \frac{-1}{2}, \frac{-9}{2} \right) = (-0.5, -4.5)
$$
✔ Answer: $ (-0.5, -4.5) $

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8) $ (9, 10), (-10, -2) $

$$
\left( \frac{9 + (-10)}{2}, \frac{10 + (-2)}{2} \right) = \left( \frac{-1}{2}, \frac{8}{2} \right) = (-0.5, 4)
$$
✔ Answer: $ (-0.5, 4) $

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9) $ \left(\frac{1}{3}, \frac{1}{8}\right), \left(0, \frac{4}{2}\right) $

Note: $ \frac{4}{2} = 2 $
$$
\left( \frac{\frac{1}{3} + 0}{2}, \frac{\frac{1}{8} + 2}{2} \right) = \left( \frac{1}{6}, \frac{\frac{17}{8}}{2} \right) = \left( \frac{1}{6}, \frac{17}{16} \right)
$$
✔ Answer: $ \left( \frac{1}{6}, \frac{17}{16} \right) $

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10) $ \left(\frac{1}{2}, \frac{4}{3}\right), \left(\frac{1}{3}, -\frac{2}{3}\right) $

$$
x: \frac{\frac{1}{2} + \frac{1}{3}}{2} = \frac{\frac{3}{6} + \frac{2}{6}}{2} = \frac{\frac{5}{6}}{2} = \frac{5}{12}
$$
$$
y: \frac{\frac{4}{3} + (-\frac{2}{3})}{2} = \frac{\frac{2}{3}}{2} = \frac{1}{3}
$$
✔ Answer: $ \left( \frac{5}{12}, \frac{1}{3} \right) $

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11) $ \left(1, -\frac{2}{3}\right), \left(1\frac{1}{7}, 3\frac{2}{3}\right) $

Convert mixed numbers:
- $ 1\frac{1}{7} = \frac{8}{7} $
- $ 3\frac{2}{3} = \frac{11}{3} $

Now:
$$
x: \frac{1 + \frac{8}{7}}{2} = \frac{\frac{15}{7}}{2} = \frac{15}{14}
$$
$$
y: \frac{-\frac{2}{3} + \frac{11}{3}}{2} = \frac{\frac{9}{3}}{2} = \frac{3}{2}
$$
✔ Answer: $ \left( \frac{15}{14}, \frac{3}{2} \right) $

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12) $ \left(-\frac{4}{3}, -1\frac{1}{2}\right), \left(\frac{1}{6}, 4\frac{1}{3}\right) $

Convert:
- $ -1\frac{1}{2} = -\frac{3}{2} $
- $ 4\frac{1}{3} = \frac{13}{3} $

Now:
$$
x: \frac{-\frac{4}{3} + \frac{1}{6}}{2} = \frac{-\frac{8}{6} + \frac{1}{6}}{2} = \frac{-\frac{7}{6}}{2} = -\frac{7}{12}
$$
$$
y: \frac{-\frac{3}{2} + \frac{13}{3}}{2} = \frac{-\frac{9}{6} + \frac{26}{6}}{2} = \frac{\frac{17}{6}}{2} = \frac{17}{12}
$$
✔ Answer: $ \left( -\frac{7}{12}, \frac{17}{12} \right) $

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13) $ \left(3\frac{1}{3}, -\frac{7}{8}\right), \left(-2, -9\frac{1}{8}\right) $

Convert:
- $ 3\frac{1}{3} = \frac{10}{3} $
- $ -9\frac{1}{8} = -\frac{73}{8} $

Now:
$$
x: \frac{\frac{10}{3} + (-2)}{2} = \frac{\frac{10}{3} - \frac{6}{3}}{2} = \frac{\frac{4}{3}}{2} = \frac{2}{3}
$$
$$
y: \frac{-\frac{7}{8} + (-\frac{73}{8})}{2} = \frac{-\frac{80}{8}}{2} = \frac{-10}{2} = -5
$$
✔ Answer: $ \left( \frac{2}{3}, -5 \right) $

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14) $ (9, -1), \left(\frac{2}{3}, 3\frac{1}{3}\right) $

Convert: $ 3\frac{1}{3} = \frac{10}{3} $

$$
x: \frac{9 + \frac{2}{3}}{2} = \frac{\frac{27}{3} + \frac{2}{3}}{2} = \frac{\frac{29}{3}}{2} = \frac{29}{6}
$$
$$
y: \frac{-1 + \frac{10}{3}}{2} = \frac{-\frac{3}{3} + \frac{10}{3}}{2} = \frac{\frac{7}{3}}{2} = \frac{7}{6}
$$
✔ Answer: $ \left( \frac{29}{6}, \frac{7}{6} \right) $

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## ✔ Part 2: Find the midpoint of each line segment from graphs.

Since we don’t have the actual images, I’ll explain how to do it.

You need to:
1. Identify the coordinates of the two endpoints from the graph.
2. Apply the midpoint formula.

But since you can't see the image, here’s a general approach:

> For example, if one endpoint is at $ (2, 4) $ and the other at $ (-2, -2) $, then:
>
> $$
> \left( \frac{2 + (-2)}{2}, \frac{4 + (-2)}{2} \right) = (0, 1)
> $$

So for #15 and #16, you would:
- Read the coordinates from the grid.
- Plug into midpoint formula.

👉 If you can describe the points on the graph, I can help compute them.

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## ✔ Part 3: Find the other endpoint given one endpoint and the midpoint.

Use the midpoint formula in reverse.

Let:
- Endpoint: $ (x_1, y_1) $
- Midpoint: $ (m_x, m_y) $
- Other endpoint: $ (x_2, y_2) $

Then:
$$
m_x = \frac{x_1 + x_2}{2} \Rightarrow x_2 = 2m_x - x_1
$$
$$
m_y = \frac{y_1 + y_2}{2} \Rightarrow y_2 = 2m_y - y_1
$$

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17) Endpoint: $(-1, 2)$, midpoint: $(0, -4)$

$$
x_2 = 2(0) - (-1) = 0 + 1 = 1 \\
y_2 = 2(-4) - 2 = -8 - 2 = -10
$$
✔ Answer: $ (1, -10) $

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18) Endpoint: $ (5, 8) $, midpoint: $ (-1, 3) $

$$
x_2 = 2(-1) - 5 = -2 - 5 = -7 \\
y_2 = 2(3) - 8 = 6 - 8 = -2
$$
✔ Answer: $ (-7, -2) $

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19) Endpoint: $ (-8, 9) $, midpoint: $ (0, -4) $

$$
x_2 = 2(0) - (-8) = 0 + 8 = 8 \\
y_2 = 2(-4) - 9 = -8 - 9 = -17
$$
✔ Answer: $ (8, -17) $

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20) Endpoint: $ (2, -10) $, midpoint: $ (6, -7) $

$$
x_2 = 2(6) - 2 = 12 - 2 = 10 \\
y_2 = 2(-7) - (-10) = -14 + 10 = -4
$$
✔ Answer: $ (10, -4) $

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21) Endpoint: $ (3, 7) $, midpoint: $ (-5, -1) $

$$
x_2 = 2(-5) - 3 = -10 - 3 = -13 \\
y_2 = 2(-1) - 7 = -2 - 7 = -9
$$
✔ Answer: $ (-13, -9) $

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22) Endpoint: $ (10, 0) $, midpoint: $ (7, 7) $

$$
x_2 = 2(7) - 10 = 14 - 10 = 4 \\
y_2 = 2(7) - 0 = 14 - 0 = 14
$$
✔ Answer: $ (4, 14) $

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## ✔ Final Answers Summary

Midpoints (Problems 1–14):

| Problem | Answer |
|--------|--------|
| 1 | $ (2.5, -4) $ |
| 2 | $ (1, 7) $ |
| 3 | $ (4.5, -1) $ |
| 4 | $ (-3.5, -0.5) $ |
| 5 | $ (-6.5, 7.5) $ |
| 6 | $ (-8, -8) $ |
| 7 | $ (-0.5, -4.5) $ |
| 8 | $ (-0.5, 4) $ |
| 9 | $ \left( \frac{1}{6}, \frac{17}{16} \right) $ |
| 10 | $ \left( \frac{5}{12}, \frac{1}{3} \right) $ |
| 11 | $ \left( \frac{15}{14}, \frac{3}{2} \right) $ |
| 12 | $ \left( -\frac{7}{12}, \frac{17}{12} \right) $ |
| 13 | $ \left( \frac{2}{3}, -5 \right) $ |
| 14 | $ \left( \frac{29}{6}, \frac{7}{6} \right) $ |

Midpoints from Graphs (15 & 16):

👉 Need graph details to solve. Please describe the endpoints.

Other Endpoints (17–22):

| Problem | Answer |
|--------|--------|
| 17 | $ (1, -10) $ |
| 18 | $ (-7, -2) $ |
| 19 | $ (8, -17) $ |
| 20 | $ (10, -4) $ |
| 21 | $ (-13, -9) $ |
| 22 | $ (4, 14) $ |

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If you can provide the coordinates from the graphs (for #15 and #16), I’ll finish those too!

Let me know!