In the given triangle $IGK$, we have the following information:

• $H$ is the midpoint of $IG$.
• $J$ is the midpoint of $IK$.
• $IG = GK = 10$.

We are comparing two quantities:

• Quantity A: $IJ$, the length between points $I$ and $J$.
• Quantity B: 5.

Step 1: Analyze the lengths

Since $H$ is the midpoint of $IG$, it means: $IH = \frac{IG}{2} = \frac{10}{2} = 5.$

Similarly, since $J$ is the midpoint of $IK$, it follows that: $IJ = \frac{IK}{2}.$

Now, from the symmetry of the triangle, we know that $IG = GK = 10$. Therefore, $IK = IG + GK = 10 + 10 = 20$.

Step 2: Calculate $IJ$

Since $J$ is the midpoint of $IK$, we have: $IJ = \frac{IK}{2} = \frac{20}{2} = 10.$

Step 3: Compare the quantities

• Quantity A: $IJ = 10$.
• Quantity B: 5.

Clearly, $IJ = 10$, which is greater than 5.

Thus, Quantity A is greater.

Would you like further details or have any questions?

Related Questions:

1. How would the result change if $IG$ and $GK$ were different lengths?
2. What would happen if $H$ and $J$ were not midpoints but points dividing the segments in different ratios?
3. Can we apply the midpoint theorem to find the lengths in any type of triangle?
4. How does the result change if the lengths of sides $IG$ and $GK$ are unequal?
5. Can this method be applied to non-triangular shapes to compare distances?

Tip:

When working with midpoints, remember that each segment is halved, making geometric properties and symmetry easier to analyze.