Question

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
hf is 2 units and gh is 3 units.
hf is 3 units and gh is 2 units.
hf is 4 units and gh is 2 units.
hf is 3 units and gh is 4 units.

Explanation:

Step1: Calculate the ratio of the known sides

We know that \(DF=DG + GF=15 + 5=20\).
We need to check the ratios for SSS similarity. The ratio of \(DE\) to \(GH\) and \(EF\) to \(HF\) and \(DF\) to \(GF\) should be equal.
The ratio of \(DF\) to \(GF\) is \(\frac{DF}{GF}=\frac{20}{5} = 4\).
The ratio of \(DE\) to \(GH\) and \(EH+HF\) to \(HF\) should also be \(4\).

Step2: Check each option

• Option 1: If \(HF = 2\) and \(GH=3\), \(\frac{DE}{GH}=\frac{8}{3}

eq4\)

• Option 2: If \(HF = 3\) and \(GH = 2\), \(\frac{DE}{GH}=\frac{8}{2}=4\) and \(\frac{EH + HF}{HF}=\frac{12+3}{3}=\frac{15}{3}=5

eq4\)

• Option 3: If \(HF = 4\) and \(GH = 2\), \(\frac{DE}{GH}=\frac{8}{2}=4\) and \(\frac{EH+HF}{HF}=\frac{12 + 4}{4}=\frac{16}{4}=4\)
• Option 4: If \(HF = 3\) and \(GH = 4\), \(\frac{DE}{GH}=\frac{8}{4}=2

eq4\)

Answer:

HF is 4 units and GH is 2 units.