Question : In $\triangle ABC$, D and E are points on the sides AB and AC, respectively, such that DE || BC. If AD = 5 cm, DB = 9 cm, AE = 4 cm, and BC = 15.4 cm, then the sum of the lengths of DE and EC (in cm) is:
Option 1: 11.6
Option 2: 10.8
Option 3: 13.4
Option 4: 12.7
Correct Answer: 12.7
Solution :
According to the question
DE || BC
using basic proportionality theorem
⇒ $\frac{AD}{DB}$ = $\frac{AE}{EC}$
⇒ $\frac{5}{9}$ = $\frac{4}{EC}$
⇒ EC = $\frac{36}{5}$ = 7.2 cm
Now, since the two triangles ADE and ABC are similar
⇒ $\frac{AD}{AB}$ = $\frac{DE}{BC}$
⇒ $\frac{5}{14}$ = $\frac{DE}{15.4}$
⇒ $DE$ = 5.5 cm
⇒ $DE + EC$ = 5.5 + 7.2 = 12.7 cm
Hence, the correct answer is 12.7.
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