Question

D, E, F are the mid-points of the sides BC, CA and AB respectively of a $△$ ABC. The ratio of the areas of $△$ ABC and $△$ DEF.

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Answer 1

Since D and E are the mid-point of the sides BC and AB respectively of $△$ ABC. Therefore,

DE || BA

$\Rightarrow$ DE || FA .......(i)

Since D and F are mid-points of the sides BC and AB respectively of $△$ ABC, Therefore,
DF I I CA 12 DF I I AE
From (I), and (ii), we conclude that AFDE is a parallelogram.

Similarly, BDEF is a parallelogram

. Nov, in ADEF and AABC, we have

$\angle$FDE = $\angle$A

[Opposite angles of parallelogram AFDE]

And, $\angle$DEF = $\angle$B

[Opposite angles of parallelogram BDEF]

So, by AA-similarity criterion, we have

$\angle$ DEF ~ $\angle$ ABC
$\frac{areaof△DEF}{areaof△ABC}$ = $\frac{D{E}^2}{A{B}^2}$ = $\frac{\frac{1}{4}A{B}^2}{A{B}^2}$
Hence,

Area of $△$ ABC: Area of $△$ DEF = 4: 1