D, E and F are respectively the mid-points of sides AB, BC and CA of Δ ABC. Find the ratio of the ar

For Competitive Exams

Question 5 D, E and F are respectively the mid-points of sides AB, BC and CA of Δ ABC. Find the ratio of the areas of Δ DEF and Δ ABC.

Solution:

D, E, and F are the mid-points of ΔABC

DE || AC and

DE =$ \dfrac{1}{2}$ AC (Midpoint theorem) …. (1)

In ΔBED and ΔBCA

$\angle BED = \angle BCA$ (Corresponding angles)

$\angle BDE = \angle BAC$ (Corresponding angles)

$\angle EBD = \angle CBA$ (Common angles)

ΔBED∼ΔBCA (AAA similarity criterion)

$\dfrac{ar (ΔBED) }{ ar (ΔBCA)}=(\dfrac{DE}{AC})^2$

$\dfrac{ar (ΔBED) }{ ar (ΔBCA)} = \dfrac{1}{4}$ [From (1)]

ar (ΔBED) = $\dfrac{1}{4}$ ar (ΔBCA)

Similarly,

ar (ΔCFE) = $\dfrac{1}{4}$ ar (CBA) and ar (ΔADF) = $\dfrac{1}{4}$ ar (ΔADF) = $\dfrac{1}{4}$ ar (ΔABC)

Also,

ar (ΔDEF) = ar (ΔABC) − [ar (ΔBED) + ar (ΔCFE) + ar (ΔADF)]

ar (ΔDEF) = ar (ΔABC) − $\dfrac{3}{4}$ ar (ΔABC) = $\dfrac{1}{4}$ ar (ΔABC)

$\dfrac{ar (ΔDEF)}{ ar (ΔABC)} =\dfrac{1}{4}$

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