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Question
In the figure , ABCD is a quadrilateral . F is a point on AD such that AF = 2.1 cm and FD = 4.9 cm . E and G are points on AC and AB respectively such that EF || CD and GE || BC . Find `("Ar" triangle "BCD")/("Ar" triangle "GEF")`
Solution
In Δ ABC, GE || BC
∴ By BPT
`"AG"/"GB" = "AF"/"FD"` .......(1)
Similarly, in Δ ACD
`"AE"/"EC" = "AF"/"FD"` .....(2)
From ( 1) and (2)
`"AG"/"GB" = "AF"/"FD"`
∴ GE || BC ...(By converse of BPT)
In Δ AGF and Δ ABD
∠ A = ∠ A (common)
∠ AFG = ∠ ADB (Corresponding angles)
∴ Δ AGF ∼ Δ ABD (AA corollary)
∴ `"AF"/"AD" = "GF"/"BD"` (Similar sides of similar triangles)
`2.1/7 = "GF"/"BD"`
`3/10 = "GF"/"BD"`
`("Ar" triangle "GEF")/("Ar" triangle "BCD") = "GF"^2/"BD"^2`
[The ration of areas of two similar triangle is equal to the ratio of square of their corresponding sides.]
= `(3/10)^2`
= `9/100`
= 9 : 100
`("Ar" triangle "BCD") : ("Ar" triangle "GEF")` = 100 : 9
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