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In this problem, we are given a triangle \( \triangle IGK \) with midpoints \( H \) and \( J \), where:

- \( H \) is the midpoint of \( IG \),
- \( J \) is the midpoint of \( IK \),
- \( IG = GK = 10 \).

We are asked to compare two quantities:

- **Quantity A**: \( \frac{IJ}{IJ} \),
- **Quantity B**: 5.

### Step-by-Step Analysis:

1. **Midpoints**: 
   - \( H \) is the midpoint of \( IG \), so \( IH = HG = \frac{IG}{2} = \frac{10}{2} = 5 \).
   - \( J \) is the midpoint of \( IK \), so \( IJ = JK = \frac{IK}{2} \).

2. **Length of \( IK \)**:
   - Since \( I \), \( G \), and \( K \) form a triangle, and \( IG = GK = 10 \), it is likely that \( IK = IG + GK = 10 + 10 = 20 \).

3. **Length of \( IJ \)**:
   - Since \( J \) is the midpoint of \( IK \), \( IJ = \frac{IK}{2} = \frac{20}{2} = 10 \).

### Comparison of Quantities:

- **Quantity A** is \( \frac{IJ}{1} = IJ = 10 \),
- **Quantity B** is 5.

### Conclusion:
- \( IJ = 10 \), which is greater than 5. Therefore, **Quantity A** is greater than **Quantity B**.

Do you have any questions or would you like further clarification? 

Here are 5 related questions to consider:
1. What is the general formula for finding the length of a segment if you know its midpoint?
2. How would the result change if \( IG \) and \( GK \) were unequal?
3. How does the midpoint theorem apply to this problem?
4. How do you calculate the perimeter of triangle \( IGK \) given the side lengths?
5. What is the significance of the midpoints in dividing a triangle?

**Tip**: Always check if a problem involves midpoint relations, as they can simplify the calculations using halved distances.

In a triangle IGK, H is the midpoint of IG, and J is the midpoint of IK. Given that IG = GK = 10, compare the following quantities:

Quantity A: Length of segment IJ
Quantity B: 5.

Explore how midpoints affect segment lengths in triangle IGK, where IG = GK = 10. Solve for the length of segment IJ and compare it to 5 using the midpoint theorem. This problem covers essential geometry concepts for students in Grades 8-10.

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