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We are given a geometric figure and some information about points and midpoints. Let's break it down:

1. **Givens**:
   - \( H \) is the midpoint of \( IG \).
   - \( J \) is the midpoint of \( IK \).
   - \( IG = GK = 10 \).

2. **Analysis**:
   - Since \( H \) is the midpoint of \( IG \), the length of \( IH = HG = 5 \) (half of \( IG \)).
   - Similarly, \( J \) is the midpoint of \( IK \), but we don't yet know \( IK \). However, we can calculate the length of \( IJ \) based on the relationship between the points.

3. **Focus on \( IJ \)**:
   - By the Midpoint Theorem (in a triangle, the segment joining the midpoints of two sides is parallel to the third side and half as long), \( IJ \) is half the length of \( GK \), which is 10.
   - Therefore, \( IJ = \frac{1}{2} \times 10 = 5 \).

4. **Comparing Quantities**:
   - **Quantity A** is \( IJ \), which is 5.
   - **Quantity B** is given as 5.

Since \( IJ = 5 \) and Quantity B is also 5, **the two quantities are equal**.

### Correct answer:  
- \( \boxed{\text{The two quantities are equal.}} \)

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What is the Midpoint Theorem, and how is it applied?
2. How would the answer change if the value of \( GK \) were different?
3. What if \( H \) were not the midpoint of \( IG \)? How would it affect the solution?
4. Can the length of \( IK \) be determined from the given information?
5. What other properties of triangles and midpoints can be useful in geometry problems?

**Tip**: Always check if midpoints create relationships like parallel lines or proportional segments in triangles, as they often simplify the problem.

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

Learn how to compare the length of a segment joining midpoints in a triangle using the Midpoint Theorem. This problem covers key geometric concepts such as midpoints and proportional segments in triangles, suitable for students in Grades 9-10.

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