Q. 164.3( 12 Votes )

If D, E, F are the mid-points of sides BC, CA and AB respectively of A ABC, then the ratio of the areas of triangles DEF and ABC isA. 1 :4B. 1 : 2C. 2 : 3D. 4 : 5

Given D, E and F are the mid-points of sides BC, CA and AB respectively of ΔABC.

Then DE || AB, DE || FA … (1)

And DF || CA, DF || AE … (2)

From (1) and (2), we get AFDE is a parallelogram.

Similarly, BDEF is a parallelogram.

In ΔADE and ΔABC,

FDE = A [Opposite angles of ||gm AFDE]

DEF = B [Opposite angles of ||gm BDEF]

By AA similarity criterion, ΔABC ~ ΔDEF.

We know that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

ar (ΔDEF): ar (ΔABC) = 1: 4

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