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: Recall SSS Similarity

For \(\triangle DFE \sim \triangle GFH\) by SSS, \(\frac{DF}{GF}=\frac{FE}{FH}=\frac{DE}{GH}\). First, find \(DF = DG + GF = 15 + 5 = 20\), \(FE = EH + HF = 12 + HF\).

Step2: Check Ratios

We know \(DE = 8\), \(GF = 5\), \(DF = 20\). So \(\frac{DF}{GF}=\frac{20}{5}=4\), \(\frac{DE}{GH}=\frac{8}{GH}\), \(\frac{FE}{FH}=\frac{12 + HF}{FH}\). For similarity, \(\frac{8}{GH}=4\) (so \(GH = 2\)) and \(\frac{12 + HF}{FH}=4\) (simplify: \(12 + HF = 4FH \Rightarrow 12 = 3FH \Rightarrow FH = 4\)? Wait, no—wait, let's check each option. Wait, maybe I miscalculated. Wait, \(DF = DG + GF = 15 + 5 = 20\), \(GF = 5\), so ratio \(DF/GF = 20/5 = 4\). Then \(DE/GH\) should be 4, so \(8/GH = 4 \Rightarrow GH = 2\). Then \(FE = EH + HF = 12 + HF\), and \(FE/FH\) should be 4, so \((12 + HF)/FH = 4 \Rightarrow 12 + HF = 4FH \Rightarrow 12 = 3FH \Rightarrow FH = 4\)? Wait, no, the options: let's check each option.

Wait, maybe I messed up. Let's re-express. The sides of \(\triangle DFE\) are \(DF = 20\), \(DE = 8\), \(FE = 12 + HF\). The sides of \(\triangle GFH\) are \(GF = 5\), \(GH\), \(FH\). For SSS similarity, the ratios of corresponding sides must be equal. So \(\frac{DF}{GF}=\frac{DE}{GH}=\frac{FE}{FH}\).

\(\frac{DF}{GF}=\frac{20}{5}=4\). So \(\frac{DE}{GH}=4 \Rightarrow \frac{8}{GH}=4 \Rightarrow GH = 2\). Then \(\frac{FE}{FH}=4 \Rightarrow \frac{12 + HF}{FH}=4 \Rightarrow 12 + HF = 4FH \Rightarrow 12 = 3FH \Rightarrow FH = 4\)? But the options: let's check the options. Wait, the options are:

1. HF=2, GH=3: \(\frac{8}{3} eq4\), no.

2. HF=3, GH=2: \(\frac{8}{2}=4\), good. Then \(FE = 12 + 3 = 15\), \(FH = 3\), \(\frac{15}{3}=5\)? Wait, no, that's not 4. Wait, I must have mixed up the sides. Wait, maybe the correspondence is \(\triangle DFE \sim \triangle GFH\), so \(DF\) corresponds to \(GF\), \(FE\) corresponds to \(FH\), \(DE\) corresponds to \(GH\). Wait, no, maybe the order is different. Wait, \(D\) corresponds to \(G\), \(F\) corresponds to \(F\), \(E\) corresponds to \(H\). So sides: \(DF\) (D to F), \(GF\) (G to F); \(FE\) (F to E), \(FH\) (F to H); \(DE\) (D to E), \(GH\) (G to H). So \(DF = 20\), \(GF = 5\); \(FE = 12 + HF\), \(FH = HF\); \(DE = 8\), \(GH = GH\). So ratios: \(DF/GF = 20/5 = 4\), \(DE/GH = 8/GH\), \(FE/FH = (12 + HF)/HF\). For similarity, all ratios must be 4. So \(8/GH = 4 \Rightarrow GH = 2\), and \((12 + HF)/HF = 4 \Rightarrow 12 + HF = 4HF \Rightarrow 12 = 3HF \Rightarrow HF = 4\). But the options: wait, the fourth option is HF=3, GH=4: \(\frac{8}{4}=2 eq4\). Wait, maybe I got the correspondence wrong. Maybe \(\triangle DFE \sim \triangle GFH\) with \(D\) to \(G\), \(E\) to \(F\), \(F\) to \(H\)? No, that doesn't make sense. Wait, let's look at the diagram: \(DE\) and \(GH\) are both vertical (assuming), \(EH\) and \(HF\) are horizontal, \(DF\) and \(GF\) are the hypotenuses. So \(\triangle DFE\) has legs \(DE = 8\), \(FE = 12 + HF\), hypotenuse \(DF = 20\). \(\triangle GFH\) has legs \(GH\), \(FH\), hypotenuse \(GF = 5\). So for similarity, the ratio of legs should be equal to ratio of hypotenuses. So \(\frac{DE}{GH}=\frac{FE}{FH}=\frac{DF}{GF}\). \(DF/GF = 20/5 = 4\). So \(\frac{DE}{GH}=4 \Rightarrow GH = 8/4 = 2\). \(\frac{FE}{FH}=4 \Rightarrow (12 + FH)/FH = 4 \Rightarrow 12 + FH = 4FH \Rightarrow 3FH = 12 \Rightarrow FH = 4\). Wait, but the options: the third option is HF=4, GH=2. Wait, the third option is "HF is 4 units and GH is 2 units"—wait, no, the options are:

- 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.

Ah! The third option is HF=4, GH=2. Let's check: \(DE = 8\), \(GH = 2\), ratio 8/2=4. \(DF = 20\), \(GF = 5\), ratio 20/5=4. \(FE = 12 + 4 = 16\), \(FH = 4\), ratio 16/4=4. So all ratios are 4, so SSS similarity holds. Wait, but the options: wait, the third option is "HF is 4 units and GH is 2 units"—but let's check the options again. Wait, the user's options:

1. HF is 2 units and GH is 3 units.

2. HF is 3 units and GH is 2 units.

3. HF is 4 units and GH is 2 units.

4. HF is 3 units and GH is 4 units.

Wait, maybe I made a mistake earlier. Let's recalculate. \(DF = DG + GF = 15 + 5 = 20\). \(GF = 5\), so ratio \(DF/GF = 20/5 = 4\). \(DE = 8\), so \(DE/GH\) should be 4, so \(GH = 8/4 = 2\). \(FE = EH + HF = 12 + HF\), so \(FE/FH\) should be 4, so \(12 + HF = 4 \times HF \Rightarrow 12 = 3 \times HF \Rightarrow HF = 4\). So the third option (HF=4, GH=2) is correct. Wait, but the options: the third option is "HF is 4 units and GH is 2 units"—yes, that's the third option. Wait, but let's check the options again. Wait, the user's options:

- First option: HF=2, GH=3: \(DE/GH = 8/3 eq 4\), no.

- Second: HF=3, GH=2: \(DE/GH = 8/2 = 4\), good. Then \(FE = 12 + 3 = 15\), \(FH = 3\), \(FE/FH = 15/3 = 5 eq 4\), so no.

- Third: HF=4, GH=2: \(DE/GH = 8/2 = 4\), \(FE = 12 + 4 = 16\), \(FH = 4\), \(FE/FH = 16/4 = 4\), \(DF/GF = 20/5 = 4\). So all ratios 4, so SSS holds.

- Fourth: HF=3, GH=4: \(DE/GH = 8/4 = 2 eq 4\), no.

Wait, so the correct option is the third one: "HF is 4 units and GH is 2 units"? But wait, the options as written: "HF is 4 units and GH is 2 units"—yes, that's the third option. Wait, but maybe I misread the options. Let me check again. The options are:

1. HF is 2 units and GH is 3 units.

2. HF is 3 units and GH is 2 units.

3. HF is 4 units and GH is 2 units.

4. HF is 3 units and GH is 4 units.

Yes, so the third option is correct. Wait, but earlier when I thought the second option, I made a mistake in \(FE/FH\). So the correct answer is the third option? Wait, no—wait, \(FE = EH + HF = 12 + HF\), and \(FH\) is the length from H to F. So if HF=4, then FE=16, FH=4, ratio 16/4=4. DE=8, GH=2, ratio 8/2=4. DF=20, GF=5, ratio 20/5=4. So all three ratios are equal (4), so SSS similarity holds. So the correct option is the third one: "HF is 4 units and GH is 2 units". Wait, but let's check the options again. Wait, the user's options:

Wait, the fourth option is "HF is 3 units and GH is 4 units"—no. Wait, maybe I messed up the correspondence. Maybe the triangles are \(\triangle DFE\) and \(\triangle GFH\), so \(D\) to \(G\), \(F\) to \(F\), \(E\) to \(H\). So sides:

- \(DF\) (D to F): 20

- \(GF\) (G to F): 5

- \(DE\) (D to E): 8

- \(GH\) (G to H):?

- \(FE\) (F to E): 12 + HF

- \(FH\) (F to H): HF

So the ratios should be \(DF/GF = DE/GH = FE/FH\).

\(DF/GF = 20/5 = 4\).

So \(DE/GH = 4 \Rightarrow 8/GH = 4 \Rightarrow GH = 2\).

\(FE/FH = 4 \Rightarrow (12 + HF)/HF = 4 \Rightarrow 12 + HF = 4HF \Rightarrow 3HF = 12 \Rightarrow HF = 4\).

So yes, HF=4, GH=2, which is the third option.

Answer:

The correct option is: HF is 4 units and GH is 2 units. (The third option)