1. **Stating the problem:** We are given two triangles, $\triangle HOT$ and $\triangle PIG$, which are similar ($\triangle HOT \sim \triangle PIG$). We want to find the relationships between the sides labeled $IP$, $IG$, and $PG$ using the properties of similar triangles. 2. **Formula and rules:** For similar triangles, corresponding sides are proportional. This means: $$\frac{HO}{PI} = \frac{OT}{IG} = \frac{HT}{PG}$$ where $HO, OT, HT$ are sides of $\triangle HOT$ and $PI, IG, PG$ are corresponding sides of $\triangle PIG$. 3. **Identify corresponding sides:** Since $\triangle HOT \sim \triangle PIG$, the vertices correspond as $H \leftrightarrow P$, $O \leftrightarrow I$, and $T \leftrightarrow G$. 4. **Set up proportions:** Using the correspondence: - $HO$ corresponds to $PI$ - $OT$ corresponds to $IG$ - $HT$ corresponds to $PG$ So the ratios are: $$\frac{HO}{PI} = \frac{OT}{IG} = \frac{HT}{PG}$$ 5. **Using the boxes:** The boxes labeled $IP$, $IG$, and $PG$ correspond to sides in $\triangle PIG$. We can express each in terms of the sides of $\triangle HOT$ and the scale factor between the triangles. 6. **Conclusion:** The lengths $IP$, $IG$, and $PG$ are proportional to $HO$, $OT$, and $HT$ respectively by the same scale factor. If the scale factor is $k = \frac{PI}{HO}$, then: $$IP = k \cdot HO, \quad IG = k \cdot OT, \quad PG = k \cdot HT$$ This completes the relationship between the sides of the two similar triangles.