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.