Midpoint formula-missing-endpoint | PDF - Free Printable

Let's solve each problem on the Midpoint Formula Worksheet step by step.

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🔷 Midpoint Formula Review

The midpoint $ M $ of a line segment with endpoints $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ is:

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

We are given:
- One endpoint
- The midpoint

We need to find the missing endpoint, say $ (x, y) $.

Let’s denote:
- Given endpoint: $ (x_1, y_1) $
- Midpoint: $ (m_x, m_y) $
- Missing endpoint: $ (x_2, y_2) $

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

We'll use this formula for each problem.

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✔ Problem 1:

Endpoint: $ (2, 5) $, Midpoint: $ (3, 4) $

Find missing endpoint $ (x, y) $:

$$
x = 2(3) - 2 = 6 - 2 = 4 \\
y = 2(4) - 5 = 8 - 5 = 3
$$

✔ Missing endpoint: $ (4, 3) $

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✔ Problem 2:

Endpoint: $ (7, 9) $, Midpoint: $ (6, 7) $

$$
x = 2(6) - 7 = 12 - 7 = 5 \\
y = 2(7) - 9 = 14 - 9 = 5
$$

✔ Missing endpoint: $ (5, 5) $

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✔ Problem 3:

Endpoint: $ (-4, 8) $, Midpoint: $ (0, 8) $

$$
x = 2(0) - (-4) = 0 + 4 = 4 \\
y = 2(8) - 8 = 16 - 8 = 8
$$

✔ Missing endpoint: $ (4, 8) $

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✔ Problem 4:

Endpoint: $ (-1, -1) $, Midpoint: $ (-1, -1) $

$$
x = 2(-1) - (-1) = -2 + 1 = -1 \\
y = 2(-1) - (-1) = -2 + 1 = -1
$$

✔ Missing endpoint: $ (-1, -1) $

> This means both endpoints are the same — it's a degenerate segment (a point).

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✔ Problem 5:

Endpoint: $ (3, 10) $, Midpoint: $ (-2, -3) $

$$
x = 2(-2) - 3 = -4 - 3 = -7 \\
y = 2(-3) - 10 = -6 - 10 = -16
$$

✔ Missing endpoint: $ (-7, -16) $

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✔ Problem 6:

Endpoint: $ (0, 4) $, Midpoint: $ (5, 3.5) $

$$
x = 2(5) - 0 = 10 \\
y = 2(3.5) - 4 = 7 - 4 = 3
$$

✔ Missing endpoint: $ (10, 3) $

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✔ Problem 7:

Endpoint: $ (6, -2) $, Midpoint: $ (4.5, -2.5) $

$$
x = 2(4.5) - 6 = 9 - 6 = 3 \\
y = 2(-2.5) - (-2) = -5 + 2 = -3
$$

✔ Missing endpoint: $ (3, -3) $

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✔ Problem 8:

Endpoint: $ (4, 4) $, Midpoint: $ (0, 0) $

$$
x = 2(0) - 4 = -4 \\
y = 2(0) - 4 = -4
$$

✔ Missing endpoint: $ (-4, -4) $

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📝 Final Answers:

| Problem | Missing Endpoint |
|--------|------------------|
| 1 | $ (4, 3) $ |
| 2 | $ (5, 5) $ |
| 3 | $ (4, 8) $ |
| 4 | $ (-1, -1) $ |
| 5 | $ (-7, -16) $ |
| 6 | $ (10, 3) $ |
| 7 | $ (3, -3) $ |
| 8 | $ (-4, -4) $ |

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✔ All problems solved using the midpoint formula in reverse.

Let me know if you'd like this as a filled-in worksheet!