Question

In $△ABC$ , 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 ?

• A. 4
• B. 6
• C. 8
• D. 10

Answer 1

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 $△ADFand△ABC$
$\frac{AD}{AB}=\frac{AF}{AC}$
$\angle A=\angle A$ (common angle)
Therefore, $△ADF\sim △ABC$ by SAS similarity
Therefore, $\angle ADB=\angle ABCand\angle AFD=\angle ACB$
Therefore, DF||BC
Similarly EF||AB and DE||AC
$△ADF\sim △ABC$
$△DBE\sim △ABC$
$△FEC\sim △ABC$
$△EDF\sim △ABC$ $△ADF\cong △DBE$
$△ADF\cong △EDF$
$△ADF\cong △FEC$
$△DBE\cong △EDF$
$△DBE\cong △FEC$
$△EDF\cong △FEC$

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