https://file.aiquickdraw.com/mathbot/file/1729465747042pmxxi2u0.jpg

Let's analyze the problem shown in the image step by step.

### Given Information:
- \( H \) is the midpoint of \( IG \).
- \( J \) is the midpoint of \( IK \).
- \( IG = GK = 10 \), so triangle \( IGK \) is isosceles with equal sides \( IG \) and \( GK \).
  
We are asked to compare:
- **Quantity A**: \( IJ \)
- **Quantity B**: 5

### Step-by-Step Solution:
1. Since \( H \) is the midpoint of \( IG \), we know that:
   \[
   IH = \frac{IG}{2} = \frac{10}{2} = 5
   \]
   
2. Similarly, since \( J \) is the midpoint of \( IK \), we know that:
   \[
   IJ = \frac{IK}{2}
   \]

3. To find \( IK \), observe that \( IG = GK = 10 \), and since triangle \( IGK \) is isosceles, \( IK \) must also be 10.

4. Therefore:
   \[
   IJ = \frac{IK}{2} = \frac{10}{2} = 5
   \]

### Comparison of Quantities:
- **Quantity A**: \( IJ = 5 \)
- **Quantity B**: 5

Thus, **the two quantities are equal**.

The correct answer is the third option: **The two quantities are equal**.

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Would you like more details on any part of the solution?

Here are 5 related questions to consider:
1. What is the significance of midpoints in a triangle in terms of geometry?
2. How can we use the properties of isosceles triangles to solve other problems like this?
3. What is the general formula for finding the length of a line segment when midpoints are involved?
4. How would the problem change if \( IG \neq GK \)?
5. How can we verify that \( IJ \) is indeed half of \( IK \)?

**Tip:** Always look for symmetry and midpoint properties when solving problems involving triangles and midpoints.

H is the midpoint of IG. J is the midpoint of IK. IG = GK = 10. Compare Quantity A (IJ) and Quantity B (5).

Learn how to compare the segment IJ in a triangle where H and J are midpoints. Explore midpoint theorem and the properties of isosceles triangles to determine that Quantity A and Quantity B are equal. Suitable for students in Grades 7-9.

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