QQuestion
In the given figure, AC ∥ DE ∥ BF. If AC = 24 cm, EG = 8 cm, GB = 16 cm and BF = 30 cm:
(a) Prove △GED ∼ △GBF
(b) Find DE
(c) Find DB : AB.
✓Answer
(a) Since DE ∥ BF:
⇒ ∠DEG = ∠GBF (alternate interior angles)
⇒ ∠EDG = ∠GFB (alternate interior angles)
Therefore, △GED ∼ △GBF by the AA criterion.
(b) Corresponding sides of similar triangles are proportional.
⇒ \frac{DE}{BF}=\frac{EG}{GB}
⇒ \frac{DE}{30}=\frac{8}{16}
⇒ DE = \frac{30\times8}{16}
DE = 15 cm.
(c) In △BAC and △BDE:
∠BAC = ∠BDE (corresponding angles, since AC ∥ DE)
⇒ ∠ABC = ∠DBE (common angle)
Therefore, △BAC ∼ △BDE by the AA criterion.
Corresponding sides are proportional:
⇒ \frac{BD}{AB}=\frac{DE}{AC}
⇒ \frac{BD}{AB}=\frac{15}{24}
⇒ \frac{BD}{AB}=\frac{5}{8}
Therefore, DB : AB = 5 : 8.
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