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.