Question

δtuv ~ δigh. find gi. gi =

Explanation:

Step1: Identify corresponding sides

Since $\triangle TUV \sim \triangle IGH$, the ratios of corresponding sides are equal. Let's find the ratio of similarity. First, identify the corresponding sides. Side $TV = 36$ corresponds to side $IH = 6$, side $UV = 30$ corresponds to side $GH = 5$, and side $TU = 18$ corresponds to side $GI$ (which we need to find).

Step2: Find the scale factor

Calculate the scale factor by dividing the length of a side in $\triangle TUV$ by the corresponding side in $\triangle IGH$. Let's use $TV$ and $IH$: $\frac{TV}{IH}=\frac{36}{6} = 6$. Wait, no, actually, since $\triangle TUV \sim \triangle IGH$, the order of the letters matters. So $T$ corresponds to $I$, $U$ corresponds to $G$, $V$ corresponds to $H$. So side $TU$ (from $T$ to $U$) corresponds to side $IG$ (from $I$ to $G$), side $UV$ (from $U$ to $V$) corresponds to side $GH$ (from $G$ to $H$), and side $TV$ (from $T$ to $V$) corresponds to side $IH$ (from $I$ to $H$). So the ratio of $\triangle TUV$ to $\triangle IGH$ is $\frac{TU}{IG}=\frac{UV}{GH}=\frac{TV}{IH}$. Let's use $UV$ and $GH$: $\frac{UV}{GH}=\frac{30}{5}=6$. Wait, but $TV$ is 36 and $IH$ is 6, $\frac{36}{6}=6$, so the scale factor from $\triangle IGH$ to $\triangle TUV$ is 6. But we need to find $GI$ (which is $IG$). So $TU$ (18) corresponds to $IG$ (GI), so $\frac{TU}{IG}=6$? Wait, no, maybe I got the ratio reversed. Let's think: if $\triangle TUV \sim \triangle IGH$, then $\frac{TU}{IG}=\frac{UV}{GH}=\frac{TV}{IH}$. So $IG=\frac{TU}{\text{scale factor}}$. Wait, let's check with $UV$ and $GH$: $UV = 30$, $GH = 5$, so $\frac{UV}{GH}=\frac{30}{5}=6$. So the scale factor of $\triangle TUV$ to $\triangle IGH$ is 6. So that means each side of $\triangle TUV$ is 6 times the corresponding side of $\triangle IGH$. Wait, no, if $\triangle TUV$ is similar to $\triangle IGH$, then $TUV$ is the larger triangle and $IGH$ is the smaller one. So $TU$ (18) corresponds to $IG$ (GI), $UV$ (30) corresponds to $GH$ (5), $TV$ (36) corresponds to $IH$ (6). So the ratio of $\triangle TUV$ to $\triangle IGH$ is $\frac{TU}{IG}=\frac{30}{5}=\frac{36}{6}=6$. So $TU = 6 \times IG$. Therefore, $IG=\frac{TU}{6}=\frac{18}{6}=3$. Wait, that makes sense. Let's verify: if $IG = 3$, then $\frac{TU}{IG}=\frac{18}{3}=6$, $\frac{UV}{GH}=\frac{30}{5}=6$, $\frac{TV}{IH}=\frac{36}{6}=6$. Yes, that works. So $GI = IG = 3$.

Answer:

3