Question

which three proportions could be used to determine the length ey? hint: color code. triangle 1 is similar to triangle 2. image of two triangles: triangle 1 (a,b,c) with sides 18 in (ac), 16 in (bc), 14 in (ab); triangle 2 (e,y,k) with sides 9 in (ek), 8 in (yk), ey (unknown). options: a) \\(\frac{ey}{9} = \frac{14}{16}\\), b) \\(\frac{ey}{8} = \frac{14}{16}\\), c) \\(\frac{18}{14} = \frac{ey}{8}\\), d) \\(\frac{8}{ey} = \frac{16}{14}\\), e) \\(\frac{16}{ey} = \frac{8}{14}\\), f) \\(\frac{16}{14} = \frac{8}{ey}\\), g) \\(\frac{18}{14} = \frac{ey}{9}\\)

Explanation:

Step1: Identify Similar Triangles

Triangle 1 (ABC) and Triangle 2 (EYK) are similar, so corresponding sides are proportional. Sides: \( AB = 14 \), \( BC = 16 \), \( AC = 18 \); \( EY =? \), \( YK = 8 \), \( EK = 9 \).

Step2: Check Proportions

• Option A: \( \frac{EY}{9} = \frac{14}{16} \) → Corresponding sides \( EY \) (AB) and \( 9 \) (EK), \( 14 \) (AB) and \( 16 \) (BC)? No, incorrect correspondence.
• Option B: \( \frac{EY}{8} = \frac{14}{16} \) → \( EY \) (AB) and \( 8 \) (YK), \( 14 \) (AB) and \( 16 \) (BC). Correct correspondence (AB/YK = AB/BC? Wait, no: AB corresponds to EY, BC to YK? Wait, AB=14, EY=?; BC=16, YK=8; AC=18, EK=9. So ratios: \( \frac{AB}{EY} = \frac{BC}{YK} = \frac{AC}{EK} \) or \( \frac{EY}{AB} = \frac{YK}{BC} = \frac{EK}{AC} \). Let's re-express: \( \frac{EY}{14} = \frac{8}{16} = \frac{9}{18} \). Simplify \( \frac{8}{16} = \frac{1}{2} \), \( \frac{9}{18} = \frac{1}{2} \). So \( \frac{EY}{14} = \frac{1}{2} \) → \( EY = 7 \). Now check options:
• Option A: \( \frac{EY}{9} = \frac{14}{16} \) → \( EY = \frac{14 \times 9}{16} = \frac{126}{16} = 7.875 \) (incorrect).
• Option B: \( \frac{EY}{8} = \frac{14}{16} \) → \( EY = \frac{14 \times 8}{16} = 7 \) (correct).
• Option C: \( \frac{18}{14} = \frac{EY}{8} \) → \( EY = \frac{18 \times 8}{14} \approx 10.285 \) (incorrect).
• Option D: \( \frac{8}{EY} = \frac{16}{14} \) → \( EY = \frac{8 \times 14}{16} = 7 \) (correct).
• Option E: \( \frac{16}{EY} = \frac{8}{14} \) → \( EY = \frac{16 \times 14}{8} = 28 \) (incorrect).
• Option F: \( \frac{16}{14} = \frac{8}{EY} \) → \( EY = \frac{14 \times 8}{16} = 7 \) (correct).
• Option G: \( \frac{18}{14} = \frac{EY}{9} \) → \( EY = \frac{18 \times 9}{14} \approx 11.571 \) (incorrect).

Wait, re-express similarity ratios. Triangle 1: sides 14, 16, 18. Triangle 2: sides EY, 8, 9. So corresponding sides: 14 (AB) ↔ EY, 16 (BC) ↔ 8 (YK), 18 (AC) ↔ 9 (EK). So ratios: \( \frac{AB}{EY} = \frac{BC}{YK} = \frac{AC}{EK} \) → \( \frac{14}{EY} = \frac{16}{8} = \frac{18}{9} \). Simplify \( \frac{16}{8}=2 \), \( \frac{18}{9}=2 \), so \( \frac{14}{EY}=2 \) → \( EY=7 \). Now check each option:

• A: \( \frac{EY}{9} = \frac{14}{16} \) → \( EY = \frac{14×9}{16} = 7.875 \) (wrong).
• B: \( \frac{EY}{8} = \frac{14}{16} \) → \( EY = \frac{14×8}{16} = 7 \) (correct, since \( \frac{EY}{YK} = \frac{AB}{BC} \) → \( \frac{EY}{8} = \frac{14}{16} \)).
• C: \( \frac{18}{14} = \frac{EY}{8} \) → \( EY = \frac{18×8}{14} ≈10.29 \) (wrong).
• D: \( \frac{8}{EY} = \frac{16}{14} \) → \( EY = \frac{8×14}{16} =7 \) (correct, \( \frac{YK}{EY} = \frac{BC}{AB} \) → \( \frac{8}{EY} = \frac{16}{14} \)).
• E: \( \frac{16}{EY} = \frac{8}{14} \) → \( EY = \frac{16×14}{8}=28 \) (wrong).
• F: \( \frac{16}{14} = \frac{8}{EY} \) → \( EY = \frac{14×8}{16}=7 \) (correct, \( \frac{BC}{AB} = \frac{YK}{EY} \) → \( \frac{16}{14} = \frac{8}{EY} \)).
• G: \( \frac{18}{14} = \frac{EY}{9} \) → \( EY = \frac{18×9}{14}≈11.57 \) (wrong).

So correct options are B, D, F.

Answer:

B. \( \frac{EY}{8} = \frac{14}{16} \), D. \( \frac{8}{EY} = \frac{16}{14} \), F. \( \frac{16}{14} = \frac{8}{EY} \)