The problem shows a triangle $\Delta FGH$ with segment $IJ$ being a midsegment of the triangle. It provides the following information:

• $IJ = 7$
• $FH = 9$
• $GH = 13$

Key Concept:

A midsegment in a triangle connects the midpoints of two sides of the triangle, and it is parallel to the third side. Additionally, the length of the midsegment is half the length of the side to which it is parallel.

Since $IJ$ is a midsegment and is parallel to side $FG$, we know the following: $IJ = \frac{1}{2} \times FG$

From this, we can calculate the length of $FG$ as: $FG = 2 \times IJ = 2 \times 7 = 14$

Perimeter of $\Delta IJH$:

Now that we have the full lengths of the sides of triangle $\Delta IJH$:

• $IJ = 7$ (given)
• $IH = \frac{1}{2} \times FH = \frac{1}{2} \times 9 = 4.5$
• $JH = \frac{1}{2} \times GH = \frac{1}{2} \times 13 = 6.5$

Thus, the perimeter of $\Delta IJH$ is: $\text{Perimeter} = IJ + IH + JH = 7 + 4.5 + 6.5 = 18$

Final Answer:

The perimeter of $\Delta IJH$ is $18$ units.

Would you like more details on any of the steps or related concepts?

Here are 5 questions to expand on this concept:

1. How does the midsegment theorem help determine relationships between triangle sides?
2. What other properties of midsegments can be used in geometry problems?
3. Can you find the area of $\Delta IJH$ if the height of the triangle is known?
4. How would the problem change if $IJ$ were not a midsegment but a random segment inside the triangle?
5. Can you derive the perimeter of $\Delta FGH$ using the information provided?

Tip: When working with midsegments, remember they create smaller triangles that are similar to the larger triangle, maintaining proportional side lengths.