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D ,E ,F are mid points of the sides BC ,CA and AB respectively of a triangle ABC . Find the ratio of the areas of tri.DEF and tri.ABC.



Construction - Join all the midpoints with the sides AB,BC.AC
since,F and E are the mid points of sides BC and AC
therefore fc //  bc and fc = 1/2 bc. therefore FEBD is a //gm.
 then, //gm FEDB = 1/2 tri FED     _________  (equation 1)
similarly, //gm. FECD = 1/2 tri. FED    ________  (equation 2)
and //gm FAED. = 1/2 tri FED    _______  (equation 3)
therefore, //gm FEDB = //gm FECD = //gm FAED     _____  (equation 4)
now, tri. FBD + tri FED = tri FED + tri ECD = tri FED +tri FAE(from eq.4)
subtracting tri FED from the above eq.
we get, 
tri FBD = ECD = FAE
now, since all the triangles are equal therefore
tri ABC = 1/4 tri FED or DEF
now taking out the ratio of the above eq.
4 * tri ABC = 1 * tri DEF          (by cross multiplication)
therefore the ratio is 1:4  

Ihope this helped you 
pls mark brainliest
2 5 2
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