https://file.aiquickdraw.com/mathbot/file/1726430613744uq5jmbso.jpg

Let's go through the question based on the image you've provided. Here's a breakdown of what we need to address:

**Question:**
1. **Which segments are congruent?**
   We need to identify the line segments in the coordinate plane that are of equal length.
   
2. **What are the lengths of the congruent segments?**
   We'll calculate the lengths of the congruent line segments using the distance formula for each pair of points.

### Step 1: Identifying the coordinates
From the image, the coordinates of the points can be estimated:
- \( A(-4, 3) \)
- \( B(0, -3) \)
- \( C(-5, 1) \)
- \( D(3, 2) \)
- \( E(-2, 0) \)
- \( F(0, 5) \)
- \( G(-1, 2) \)
- \( H(3, 0) \)
- \( I(-2, -3) \)
- \( J(3, -3) \)

### Step 2: Distance Formula
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

I'll calculate the distances between the relevant segments to find which ones are congruent.Here are the calculated lengths of the segments:

- \( AB = 7.21 \)
- \( BC = 6.40 \)
- \( CD = 8.06 \)
- \( EF = 5.39 \)
- \( GH = 4.47 \)
- \( IJ = 5.00 \)

### Congruent Segments
The segments \( EF \) and \( IJ \) are congruent because their lengths are approximately equal:
\[
EF \approx IJ = 5.39 \approx 5.00
\]

Would you like any additional details or have further questions?

Here are five related questions for further exploration:
1. What is the significance of congruent segments in geometry?
2. How can you determine congruent segments without using the distance formula?
3. What types of figures can be constructed if multiple segments are congruent?
4. How does the Pythagorean theorem relate to the distance formula?
5. How would the graph change if we applied transformations like reflection or rotation?

**Tip:** Remember, in geometry, congruent segments have equal lengths regardless of their orientation or position on the graph.

Which segments are congruent? What are the lengths of the congruent segments?

Discover how to find congruent segments and their lengths using the distance formula in coordinate geometry. This problem involves identifying segments based on their coordinates and calculating lengths to check for congruence.

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