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D, E and F are respectively the mid-points of the sides BC, CA and AB of $Δ A B C$ . Determine the ratio of the areas of triangles DEF and ABC.

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2;4

1:4

2:3

5:6

answer is B.

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Detailed Solution

It is given that D, E and F are respectively the mid-points of the sides BC, CA and AB of

ΔABC .

From the question we can determine that the midpoints are D, E and F.
Let us draw the figure according to the given data.
Question Image

From the figure we can understand that,
To find the ratio of the areas of ΔDEF and ΔABC we need to prove if the triangles are similar or not.

Δ D E F & Δ A B C ∠ F D E = ∠ A ∠ D E F = ∠ B Δ D E F ~ Δ A B C

The above triangles are similar by AA similarity criterion.
Now, we know that the ratio of the area of similar triangles is equal to the ratio of the square of the sides. So,
Question Image

The ratio is

a r Δ D E F : a r Δ A B C = 1 : 4

.
Hence the correct option is 2.

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