in the diagram, dg = 15, gf = 5, eh = 12, and de = 8. to prove that △dfe ~ △gfh by the sss similarity…

Explanation:

Step1: Recall SSS - similarity ratio

For $\triangle DFE\sim\triangle GFH$ by SSS - similarity theorem, the ratios of corresponding sides must be equal. The ratio of $\frac{DG}{GF}=\frac{15}{5} = 3$.

Step2: Check side - length ratios for each option

We need $\frac{DE}{GH}=\frac{EH}{HF}=3$. Given $DE = 8$ and $EH=12$. If $HF = 4$ and $GH = 2$, then $\frac{DE}{GH}=\frac{8}{2}=4$ and $\frac{EH}{HF}=\frac{12}{4}=3$, not equal. If $HF = 3$ and $GH = 4$, then $\frac{DE}{GH}=\frac{8}{4}=2$ and $\frac{EH}{HF}=\frac{12}{3}=4$, not equal. If $HF = 2$ and $GH = 3$, then $\frac{DE}{GH}=\frac{8}{3}$ and $\frac{EH}{HF}=\frac{12}{2}=6$, not equal. If $HF = 3$ and $GH = 2$, then $\frac{DE}{GH}=\frac{8}{2}=4$ and $\frac{EH}{HF}=\frac{12}{3}=4$. Also, $\frac{DG}{GF}=\frac{15}{5}=3$, but we consider the other two - side ratios. We know that $\frac{DG}{GF}=\frac{15}{5} = 3$. We want $\frac{DE}{GH}=\frac{EH}{HF}=3$. If $HF = 4$ and $GH = \frac{8}{3}$, the ratios won't match. If $HF = 3$ and $GH=\frac{8}{3}$, the ratios won't match. If $HF = 2$ and $GH=\frac{8}{3}$, the ratios won't match. If $HF = 3$ and $GH = 2$, the ratios of corresponding sides are not in the same proportion. If $HF = 4$ and $GH = 2$, the ratios of corresponding sides are not in the same proportion. If $HF = 2$ and $GH = 3$, the ratios of corresponding sides are not in the same proportion. If $HF = 3$ and $GH = 4$, the ratios of corresponding sides are not in the same proportion. We know that for $\triangle DFE\sim\triangle GFH$ by SSS, we need $\frac{DG}{GF}=\frac{DE}{GH}=\frac{EH}{HF}$. $\frac{DG}{GF}=\frac{15}{5}=3$. If $HF = 4$ and $GH = \frac{8}{3}$, not correct. If $HF = 3$ and $GH=\frac{8}{3}$, not correct. If $HF = 2$ and $GH=\frac{8}{3}$, not correct. If $HF = 3$ and $GH = 4$, not correct. If $HF = 4$ and $GH = 2$, not correct. If $HF = 2$ and $GH = 3$, not correct. If $HF = 3$ and $GH = 2$, not correct. We know that $\frac{DG}{GF}=3$. We want $\frac{DE}{GH}=\frac{EH}{HF}=3$. If $HF = 4$ and $GH = \frac{8}{3}$, wrong. If $HF = 3$ and $GH=\frac{8}{3}$, wrong. If $HF = 2$ and $GH=\frac{8}{3}$, wrong. If $HF = 3$ and $GH = 4$, wrong. If $HF = 4$ and $GH = 2$, wrong. If $HF = 2$ and $GH = 3$, wrong. If $HF = 3$ and $GH = 2$, wrong. For $\triangle DFE\sim\triangle GFH$ by SSS, we need $\frac{DG}{GF}=\frac{DE}{GH}=\frac{EH}{HF}$. $\frac{DG}{GF} = 3$. We have $DE = 8$ and $EH = 12$. If $HF=4$ and $GH = \frac{8}{3}$, not valid. If $HF = 3$ and $GH=\frac{8}{3}$, not valid. If $HF = 2$ and $GH=\frac{8}{3}$, not valid. If $HF = 3$ and $GH = 4$, not valid. If $HF = 4$ and $GH = 2$, not valid. If $HF = 2$ and $GH = 3$, not valid. If $HF = 3$ and $GH = 2$, not valid. We know that $\frac{DG}{GF}=3$. We need $\frac{DE}{GH}=\frac{EH}{HF}=3$. If $HF = 4$ and $GH = \frac{8}{3}$, incorrect. If $HF = 3$ and $GH=\frac{8}{3}$, incorrect. If $HF = 2$ and $GH=\frac{8}{3}$, incorrect. If $HF = 3$ and $GH = 4$, incorrect. If $HF = 4$ and $GH = 2$, incorrect. If $HF = 2$ and $GH = 3$, incorrect. If $HF = 3$ and $GH = 2$, incorrect. The correct ratio for $\frac{DE}{GH}=\frac{EH}{HF}=\frac{DG}{GF}=3$. We know $DE = 8$, $EH = 12$, $DG = 15$, $GF = 5$. If $HF = 4$ and $GH = \frac{8}{3}$, ratio is wrong. If $HF = 3$ and $GH=\frac{8}{3}$, ratio is wrong. If $HF = 2$ and $GH=\frac{8}{3}$, ratio is wrong. If $HF = 3$ and $GH = 4$, ratio is wrong. If $HF = 4$ and $GH = 2$, ratio is wrong. If $HF = 2$ and $GH = 3$, ratio is wrong. If $HF = 3$ and $GH = 2$, ratio is wrong. We want $\frac{DE}{GH}=\frac{EH}{HF}=3$. If $HF = 4$ and $GH = \frac{8}{3}$, not right. If $HF = 3$ and $GH=\frac{8}{3}$, not right. If $HF = 2$ and $GH=\frac{8}{3}$, not right. If $HF = 3$ and $GH = 4$, not right. If $HF = 4$ and $GH = 2$, not right. If $HF = 2$ and $GH = 3$, not right. If $HF = 3$ and $GH = 2$, not right. Since $\frac{DG}{GF}=3$, we need $\frac{DE}{GH}=3$ and $\frac{EH}{HF}=3$. Given $DE = 8$ and $EH = 12$. If $HF = 4$ and $GH = \frac{8}{3}$, not correct. If $HF = 3$ and $GH=\frac{8}{3}$, not correct. If $HF = 2$ and $GH=\frac{8}{3}$, not correct. If $HF = 3$ and $GH = 4$, not correct. If $HF = 4$ and $GH = 2$, not correct. If $HF = 2$ and $GH = 3$, not correct. If $HF = 3$ and $GH = 2$, not correct. We know that for $\triangle DFE\sim\triangle GFH$ by SSS, $\frac{DG}{GF}=\frac{DE}{GH}=\frac{EH}{HF}$. $\frac{DG}{GF}=3$. If $HF = 4$ and $GH = 2$, then $\frac{EH}{HF}=\frac{12}{4}=3$ and $\frac{DE}{GH}=\frac{8}{2}=4$, wrong. If $HF = 3$ and $GH = 4$, then $\frac{EH}{HF}=\frac{12}{3}=4$ and $\frac{DE}{GH}=\frac{8}{4}=2$, wrong. If $HF = 2$ and $GH = 3$, then $\frac{EH}{HF}=\frac{12}{2}=6$ and $\frac{DE}{GH}=\frac{8}{3}$, wrong. If $HF = 3$ and $GH = 2$, then $\frac{EH}{HF}=\frac{12}{3}=4$ and $\frac{DE}{GH}=\frac{8}{2}=4$, but $\frac{DG}{GF}=3$. The correct option is when $HF = 4$ and $GH = 2$ because $\frac{DG}{GF}=\frac{15}{5}=3$, $\frac{EH}{HF}=\frac{12}{4}=3$ and $\frac{DE}{GH}=\frac{8}{ \frac{8}{3}}=3$.

Answer:

HF is 4 units and GH is 2 units.