23.
In the given figure, DE is parallel to the base BC of triangle ABC
Area of ADEF
and AD: DB = 3: 5. Find the ratio
Area of ACFB
(A) 25:9
(B) 9:25
B
C) 9:64
(D) 25:64
Answers
The ratio of the area of ∆DEF and area of ∆CFB is option (C): 9:64.
Step-by-step explanation:
It is given that,
In ∆ABC, DE // BC
AD:DB = 3:5
Step 1:
Consider ∆ADE and ∆ABC, we have
∠A = ∠A ….. [common angle]
∠ADE = ∠ABC ……. [corresponding angles]
∴ By AA similarity, ∆ADE ~ ∆ABC
Since the corresponding sides of similar triangles are proportional to each other.
∴ AD/AB = DE/BC
⇒ AD/(AD+DB) = DE/BC
⇒ 3/(3+5) = DE/BC
⇒ DE/BC = 3/8 ……. (i)
Step 2:
Consider ∆DFE and ∆CFB, we have
∠DFE = ∠BFC …… [vertically opposite angles]
∠EDF = ∠BCF ……. [alternate angles]
∴ By AA similarity, ∆DFE ~ ∆CFB
We know that the ratio of the two similar triangles is equal to the ratio of the square of their corresponding sides.
∴ [Area (∆DFE)] / [Area (∆CFB)] = [DE²]/[BC²]
Substituting the value from (i)
⇒ [Area (∆DFE)] / [Area (∆CFB)] = [3²]/[8²] = [9/64] ← option (C)
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