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प्रश्न
If D, E, F are the respectively the midpoints of sides BC, CA and AB of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC.
उत्तर
By using mid theorem i.e., the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
∴ DF || BC
ANd `DF=1/2 BC`
⟹ DF = BE
Since, the opposite sides of the quadrilateral are parallel and equal.
Hence, BDFE is a parallelogram
Similarly, DFCE is a parallelogram.
Now, in ΔABC and ΔEFD
∠𝐴𝐵𝐶= ∠𝐸𝐹𝐷 (𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚)
∠𝐵𝐶𝐴=∠𝐸𝐷𝐹 (𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚)
By AA similarity criterion, ΔABC ~ ΔEFD
If two triangles are similar, then the ratio of their areas is equal to the squares of their corresponding sides.
∴ `(area(Δ DEF)) /(area(ΔABC))=((DF)/(BC))^2=((DF)/(2DF))^2=1/4`
Hence, the ratio of the areas of ΔDEF and ΔABC is 1 : 4.
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