Question

ΔABC and DEF are equilateral triangles. If A(ΔABC):A(ΔDEF) = 1:4 and AB = 4, find DE. 


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Answer 1

Answer:

The answer will be 8 units.

Step-by-step explanation:

The things that we have given are :

                 Δ ABC and Δ DEF are equilateral triangle;

                 ar (ΔABC) : ar(ΔDEF) = 1:4;

and           AB = 4 unit;

Both the triangles are equilateral hence;

              ∠A = ∠D = 60;

              ∠B = ∠E = 60;

              ∠C = ∠F = 60;

Hence, Δ ABC ~ Δ DEF (by AAA criterion of similarity).

Now, for a similar triangle the ratio of their area will be :

             ar (ΔABC) / ar(ΔDEF) = (AB / DE)^2 = (BC / EF)^2 =(AC / DF)^2;

   Now, according to given things, we can conclude that;

     ⇒        ar (ΔABC) / ar(ΔDEF) = (AB / DE)^2

                 1 / 4 = (4 / DE)^2;        (since, ar (ΔABC) / ar(ΔDEF) = 1/4 and AB = 4 is given.)

     ⇒            1 / 4 = 16 / DE^2;

     ⇒           DE^2 = 16 * 4;

    ⇒             DE = $\sqrt{16*4}$;

    ⇒             DE = 4 * 2;

                          = 8 units;

*( ar (ΔABC) : ar(ΔDEF) = ar (ΔABC) / ar(ΔDEF).

That's all.