ΔEFG and ΔHIJ are similar. Also, ∠E = ∠H and ∠F = ∠I. If 3EF = HI and FG = 9 cm, then what is the length of IJ?

Question

Accepted Answer

Given: ΔEFG ~ ΔHIJ (similar triangles), ∠E = ∠H, ∠F = ∠I, 3EF = HI, FG = 9 cm.

Let EF = x. Then HI = 3x.

In similar triangles, the ratio of corresponding sides is constant. So, 
EF/HI = FG/IJ
=> x/3x = 9/IJ
=> 1/3 = 9/IJ
=> IJ = 9 × 3 = 27

But this seems off. Let's check the correspondence of sides. Since ∠E = ∠H and ∠F = ∠I, the correspondence is E→H, F→I, G→J. So, EF corresponds to HI, FG corresponds to IJ.

So, EF/HI = FG/IJ
Given HI = 3EF, so EF/HI = EF/(3EF) = 1/3
So, FG/IJ = 1/3
Given FG = 9 cm
So, 9/IJ = 1/3
=> IJ = 9 × 3 = 27 cm

But the answer marked correct is 2.27 cm, which suggests a decimal. Let's check the calculation:

If 3EF = HI, then HI/EF = 3, so the scale factor from ΔEFG to ΔHIJ is 3 (HI is 3 times EF). But since the triangles are similar, the ratio of corresponding sides is the same. So, FG/IJ = EF/HI = 1/3
So, FG/IJ = 1/3
FG = 9 cm
So, 9/IJ = 1/3
=> IJ = 9 × 3 = 27 cm

But this is not matching the answer. Let's check if the scale factor is the other way around (maybe 3EF = HI means EF = HI/3, so HI is smaller than EF):
If 3EF = HI, then EF = HI/3, so HI/EF = 3, so the scale factor is 3.
So, FG/IJ = 1/3
FG = 9 cm
9/IJ = 1/3
IJ = 27 cm

But the answer marked correct is 2.27 cm. Let's check if the scale factor is 1/3 instead:
If 3EF = HI, then HI = 3EF, so HI/EF = 3, so the scale factor is 3.
So, FG/IJ = 1/3
FG = 9 cm
9/IJ = 1/3
IJ = 27 cm

Alternatively, if the scale factor is 1/3 (i.e., HI = EF/3), then EF/HI = 3
So, FG/IJ = 3
9/IJ = 3
IJ = 9/3 = 3 cm

But none of these match 2.27 cm. Let's check if the answer is a decimal approximation of 9/4 = 2.25 cm. Let's try that:
Suppose the scale factor is k, so HI = k × EF
Given 3EF = HI, so HI/EF = 3, so k = 3
So, FG/IJ = 1/3
FG = 9 cm
9/IJ = 1/3
IJ = 27 cm

Alternatively, is there a typo in the question? If FG = 9 cm, and the answer is 2.27 cm, then the ratio is 9/2.27 ≈ 3.96, which is close to 4.
So, if the scale factor is 4, then HI = 4EF, so 3EF = HI is not possible.

Alternatively, perhaps the correct answer is 3 cm, but the answer key says 2.27 cm. Let's check 9/4 = 2.25 cm, which is close to 2.27 cm.

Alternatively, perhaps the question meant 3EF = HI, but the triangles are in the reverse order, so the scale factor is 1/3.
So, EF/HI = 3EF/HI = 1

Alternatively, perhaps the answer key is incorrect. Based on the standard approach:
- Corresponding sides: EF/HI = FG/IJ
- Given HI = 3EF, so EF/HI = 1/3
- FG = 9 cm
- So, 9/IJ = 1/3
- IJ = 27 cm

Therefore, the correct answer should be 27 cm,, perhaps the question intended FG = 3 cm, then IJ = 9 cm, or FG = 2.27 cm, then IJ = 9 cm.

Given the information, the correct mathematical answer is 27 cm,, based on the similarity ratio, the answer should be 27 cm.

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