Q Let A1 be the area of acute angles triangle inside which another triangle of area A2 is drawn - Maths - Straight Lines

Question

Q Let A 1 be the area of acute angles triangle inside which another
triangle of area A 2 is drawn such that it's sides are parallel to
the sides of the original triangle If the process is repeated again
to form triangles of areas A n , n=3,4 , Then find value of A 1 / A
4

64
16
8
256

Aarushi Mishra · Accepted Answer

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Consider the above figureLEt DE ||BC, EF||AB and DF||ACIn quadrilateral DEFBDE||FBEF||BDTherefore, DEFB is a prallelogramHenceEF=BD_____1Now consider quadrilateral ADFEEF||ADAE||DFTherefore, ADFE is a parallelogramHenceEF=AD ____________2From equation 1 and 2AD=BDTherefore, D is mid-point of ABSimialrly we can show that E and F are midpoints of AC and BC respectivelyResult: The area of triangle formed by joining midpoints of its sides is 14th time the area of the tirangleBack to our questionUsing the result we have provedtriangle with area A2 is the tirangle formed by joining midpoints of sides of triangle with area A1
 similarly, triangle with area A3 is the tirangle formed by joining midpoints of sides of triangle with area A2 and so on
A2=14 A1A3=14 A2=14 14 A1=142 A1A4=14 A3=14 142 A1=143 A1A4A1=143 =164A1A4=64

Q. Let A 1 be the area of acute angles triangle inside which another triangle of area A 2 is drawn such that it's sides are parallel to the sides of the original triangle . If the process is repeated again to form triangles of areas A n , n=3,4...... , Then find value of A 1 / A 4 . 64 16 8 256