In $△ A B C$ , D,E and F are mid points of sides AB , BC and AC respectively.How many pairs of similar triangles are there in the figure formed after joining D,E,F ?

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Solution

The correct option is D
10

The following is the figure in which we need to find the number of similar triangles
Let AB =2x, BC = 2y, AC = 2z In $△ A D F a n d △ A B C$
$\frac{A D}{A B} = \frac{A F}{A C}$
$∠ A = ∠ A$ (common angle)
Therefore, $△ A D F \sim △ A B C$ by SAS similarity
Therefore, $∠ A D B = ∠ A B C a n d ∠ A F D = ∠ A C B$
Therefore, DF||BC
Similarly EF||AB and DE||AC
$△ A D F \sim △ A B C$
$△ D B E \sim △ A B C$
$△ F E C \sim △ A B C$
$△ E D F \sim △ A B C$ $△ A D F ≅ △ D B E$
$△ A D F ≅ △ E D F$
$△ A D F ≅ △ F E C$
$△ D B E ≅ △ E D F$
$△ D B E ≅ △ F E C$
$△ E D F ≅ △ F E C$

Since all congruent figures are similar, there are 10 pairs of similar triangles

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