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

in the diagram, dg = 15, gf = 5, eh = 12, and de = 8. to prove that △dfe ~ △gfh by the sss similarity theorem using the information provided in the diagram, it would be enough additional information to know that\n○ hf is 2 units and gh is 3 units.\n○ hf is 3 units and gh is 2 units.\n○ hf is 4 units and gh is 2 units.\n○ hf is 3 units and gh is 4 units.

in the diagram, dg = 15, gf = 5, eh = 12, and de = 8. to prove that △dfe ~ △gfh by the sss similarity theorem using the information provided in the diagram, it would be enough additional information to know that\n○ hf is 2 units and gh is 3 units.\n○ hf is 3 units and gh is 2 units.\n○ hf is 4 units and gh is 2 units.\n○ hf is 3 units and gh is 4 units.

Answer

Explanation:

Step1: Recall SSS similarity ratio

For $\triangle DFE\sim\triangle GFH$ by SSS similarity theorem, we need $\frac{DE}{GF}=\frac{EF}{FH}=\frac{DF}{GH}$. Given $DE = 8$, $GF=5$, $DG = 15$, $EH = 12$. First, find $\frac{DE}{GF}=\frac{8}{5}$.

Step2: Calculate $EF$

$EF=EH + HF$. Let's check each option for the ratio of corresponding - sides.

Step3: Check option 1

If $HF = 2$ and $GH = 3$, then $EF=12 + 2=14$, and $\frac{EF}{FH}=\frac{14}{2}=7\neq\frac{8}{5}$.

Step4: Check option 2

If $HF = 3$ and $GH = 2$, then $EF=12 + 3 = 15$, and $\frac{EF}{FH}=\frac{15}{3}=5\neq\frac{8}{5}$.

Step5: Check option 3

If $HF = 4$ and $GH = 2$, then $EF=12 + 4=16$, and $\frac{EF}{FH}=\frac{16}{4}=4\neq\frac{8}{5}$.

Step6: Check option 4

If $HF = 3$ and $GH = 4$, then $EF=12 + 3=15$, $DF=DG + GF=15 + 5 = 20$. $\frac{DE}{GF}=\frac{8}{5}$, $\frac{EF}{FH}=\frac{15}{3}=5$, $\frac{DF}{GH}=\frac{20}{4}=5$. Since $\frac{DE}{GF}=\frac{EF}{FH}=\frac{DF}{GH}$, the triangles are similar by SSS - similarity.

Answer:

HF is 3 units and GH is 4 units.