Midpoint And Distance Formula Worksheet Pdf - Fill and Sign ... - Free Printable

It looks like you've uploaded a worksheet titled "Geometry Notes: BCL 1.2, Midpoints and Distance Formulas", which includes definitions, examples, and practice problems related to finding midpoints of segments in the coordinate plane.

Since I can't view images directly, I’ll help you solve the problems based on the text you’ve provided. Let’s go through each part step by step.

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🔹 Definitions (Review)

- Midpoint of a Segment: The point that divides a segment into two congruent parts.
- Segment Bisector: A point, ray, line, or line segment that intersects the segment at its midpoint.

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🔹 Example Problem

> Find the midpoint of a segment given endpoints A(3, -1) and B(5, 7).

We use the Midpoint Formula:

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

Plug in the coordinates:

$$
M = \left( \frac{3 + 5}{2}, \frac{-1 + 7}{2} \right) = \left( \frac{8}{2}, \frac{6}{2} \right) = (4, 3)
$$

✔ Answer: The midpoint is $ M = (4, 3) $

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🔹 Practice Problems

Let’s solve each one:

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#### ✔ a) Find the midpoint of $ \overline{AC} $ given:
- $ A(-2, 1) $
- $ C(6, 9) $

Use the midpoint formula:

$$
M = \left( \frac{-2 + 6}{2}, \frac{1 + 9}{2} \right) = \left( \frac{4}{2}, \frac{10}{2} \right) = (2, 5)
$$

✔ Answer: $ M = (2, 5) $

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#### ✔ b) Given midpoint $ M(5, 3) $ and endpoint $ B(8, 1) $, find the coordinates of the other endpoint $ A $.

Let $ A = (x, y) $. Use the midpoint formula:

$$
M = \left( \frac{x + 8}{2}, \frac{y + 1}{2} \right) = (5, 3)
$$

Set up equations:

1. $ \frac{x + 8}{2} = 5 $ → $ x + 8 = 10 $ → $ x = 2 $
2. $ \frac{y + 1}{2} = 3 $ → $ y + 1 = 6 $ → $ y = 5 $

✔ Answer: $ A = (2, 5) $

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#### ✔ c) Find the midpoint given:
- $ A(7, -2) $
- $ C(0, 8.7) $

$$
M = \left( \frac{7 + 0}{2}, \frac{-2 + 8.7}{2} \right) = \left( \frac{7}{2}, \frac{6.7}{2} \right) = (3.5, 3.35)
$$

✔ Answer: $ M = (3.5, 3.35) $

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#### ✔ d) Given midpoint $ M(6, 1) $, $ A(2, -3) $, find $ C(x, y) $

Let $ C = (x, y) $

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

Solve:

1. $ \frac{2 + x}{2} = 6 $ → $ 2 + x = 12 $ → $ x = 10 $
2. $ \frac{-3 + y}{2} = 1 $ → $ -3 + y = 2 $ → $ y = 5 $

✔ Answer: $ C = (10, 5) $

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🔹 Additional Problem (From Worksheet)

> Point M is the midpoint of $ \overline{WV} $. Find the length of $ \overline{WV} $.
>
> Given: $ W(2, 4) $, $ M(3, 1) $, $ V(x, y) $? Wait — actually, we need to find $ WV $, but only $ W $ and $ M $ are given.

Wait! Let's re-read.

Actually, the problem says:
> "Point M is the midpoint of $ \overline{WV} $. What do we know about $ \overline{WM} $ and $ \overline{MV} $? Use this fact to solve."

But then it gives:
- $ W(2, 4) $
- $ M(3, 1) $
- $ V(4, -2) $? Wait — no, let's see.

Looking at your image description:

> "What do we know about $ \overline{WM} $ and $ \overline{MV} $? Use this fact to solve."

Then it says:
> "You Try: Find the segment bisector of $ \overline{PQ} $. Then find $ PQ $"

Wait — maybe there's a diagram missing?

But then it says:

> Find the midpoint of a segment in the coordinate plane using the midpoint formula.

Then:

> Find the midpoint of a segment in the coordinate plane by finding the average (mean) of the abscissa (x-coordinate) and ordinate (y-coordinate).

And then:

> Given points $ A(x_1, y_1) $ and $ B(x_2, y_2) $, the midpoint has coordinates:
> $$
> M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
> $$

So, likely the main task is solving the four problems labeled a–d.

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✔ Summary of Answers:

| Problem | Answer |
|--------|--------|
| a) Midpoint of $ \overline{AC} $: $ A(-2,1), C(6,9) $ | $ (2, 5) $ |
| b) Endpoint $ A $? Given $ M(5,3), B(8,1) $ | $ (2, 5) $ |
| c) Midpoint of $ A(7,-2), C(0,8.7) $ | $ (3.5, 3.35) $ |
| d) Endpoint $ C $? Given $ M(6,1), A(2,-3) $ | $ (10, 5) $ |

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🔍 Extra Tip: Understanding the Midpoint

The midpoint is essentially the average of the x-coordinates and the average of the y-coordinates.

For example:
- If $ A(2, 3) $, $ B(6, 7) $, then:
- $ x $-midpoint: $ \frac{2+6}{2} = 4 $
- $ y $-midpoint: $ \frac{3+7}{2} = 5 $
- So $ M = (4, 5) $

This makes sense geometrically — it’s the center point between two ends.

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If you have any specific question about a problem, or if there was a diagram (like $ \overline{WV} $ with $ M $ as midpoint), feel free to describe it and I can help further!

Let me know if you'd like help with distance formula problems too (which often come next after midpoint).