Question

which triangles are similar to △abc?

choose 1 answer:
a △def only
b △ghi only
c both
d neither

Explanation:

Step1: Identify sides of △ABC

△ABC has sides \( AB = 8 \), \( BC = 8 \), \( AC = 11 \). So the side lengths are \( 8, 8, 11 \) (two equal sides, isosceles triangle).

Step2: Analyze △DEF

△DEF has sides \( DE = 4 \), \( EF = 4 \), \( DF = 5.5 \). Check the ratios:
\( \frac{DE}{AB} = \frac{4}{8} = \frac{1}{2} \), \( \frac{EF}{BC} = \frac{4}{8} = \frac{1}{2} \), \( \frac{DF}{AC} = \frac{5.5}{11} = \frac{1}{2} \). All ratios are equal, so △DEF ~ △ABC by SSS similarity.

Step3: Analyze △GHI

△GHI has sides \( GH = 16 \), \( HI = 16 \), \( GI = 22 \). Check the ratios:
\( \frac{GH}{AB} = \frac{16}{8} = 2 \), \( \frac{HI}{BC} = \frac{16}{8} = 2 \), \( \frac{GI}{AC} = \frac{22}{11} = 2 \). All ratios are equal, so △GHI ~ △ABC by SSS similarity. Wait, but the option for △GHI is labeled as "△GHI only" but let's recheck the problem's options. Wait, the options are: A: △DEF only, B: △GHI only, C: Both, D: Neither. Wait, in the problem, △GHI's sides: Wait, the diagram shows GI = 22, GH = 16, HI = 16? Wait, no, the labels: G to I is 22, G to H is 16, H to I is 16? Wait, no, the triangle is GHI, with GI = 22, GH = 16, HI = 16? Wait, no, the sides: AB = 8, BC = 8, AC = 11. △DEF: DE = 4, EF = 4, DF = 5.5. Ratios: 4/8 = 0.5, 4/8 = 0.5, 5.5/11 = 0.5. So similar. △GHI: GH = 16, HI = 16, GI = 22. Ratios: 16/8 = 2, 16/8 = 2, 22/11 = 2. So also similar. But the options: Wait, the problem's option B is "△GHI only", but according to calculation, both are similar. Wait, maybe I misread the triangle labels. Wait, the original △ABC: AB = 8, BC = 8, AC = 11. △DEF: DE = 4, EF = 4, DF = 5.5. So DE corresponds to AB, EF to BC, DF to AC. Ratios 4/8 = 1/2, 4/8 = 1/2, 5.5/11 = 1/2. So similar. △GHI: GH = 16, HI = 16, GI = 22. GH corresponds to AB (8), HI to BC (8), GI to AC (11). 16/8 = 2, 16/8 = 2, 22/11 = 2. So also similar. But the options are A: △DEF only, B: △GHI only, C: Both, D: Neither. So the correct answer is C: Both? Wait, but the user's problem shows the options as:

A: △DEF only

B: △GHI only

C: Both

D: Neither

Wait, maybe I made a mistake. Wait, let's re-express the sides:

△ABC: sides 8, 8, 11 (isosceles, two sides 8, one 11)

△DEF: sides 4, 4, 5.5 (two sides 4, one 5.5) → ratios 4/8 = 0.5, 4/8 = 0.5, 5.5/11 = 0.5 → similar.

△GHI: sides 16, 16, 22 (two sides 16, one 22) → ratios 16/8 = 2, 16/8 = 2, 22/11 = 2 → similar.

Thus, both △DEF and △GHI are similar to △ABC. So the answer is C: Both.

Answer:

C. Both