1. **State the problem:** We have triangle $\triangle FGH$ with $IJ$ as a midsegment parallel to side $FG$. Given $IJ=11$, $FH=17$, and $GH=21$, find the perimeter of $\triangle IJH$. 2. **Recall the midsegment theorem:** A midsegment in a triangle connects the midpoints of two sides and is parallel to the third side. Its length is half the length of the side it is parallel to. 3. **Apply the theorem:** Since $IJ$ is a midsegment parallel to $FG$, then: $$IJ = \frac{1}{2} FG$$ Given $IJ=11$, solve for $FG$: $$FG = 2 \times 11 = 22$$ 4. **Find the perimeter of $\triangle IJH$:** The triangle $\triangle IJH$ has sides $IJ=11$, $IH$, and $JH$. 5. **Find lengths $IH$ and $JH$:** Since $I$ and $J$ are midpoints of $FH$ and $GH$ respectively, segments $IH$ and $JH$ are half of $FH$ and $GH$ respectively: $$IH = \frac{1}{2} FH = \frac{1}{2} \times 17 = 8.5$$ $$JH = \frac{1}{2} GH = \frac{1}{2} \times 21 = 10.5$$ 6. **Calculate the perimeter:** $$\text{Perimeter} = IJ + IH + JH = 11 + 8.5 + 10.5 = 30$$ **Final answer:** The perimeter of $\triangle IJH$ is $30$.