in a triangle abc the medians am and cn to sides bc and ab intersect in point O. P is the mid point of side AC and MP intersect CN in Q. if the area of triangle OMQ is n then find area of triangle ABC
Answers
Given : in a triangle abc the medians am and cn to sides bc and ab intersect in point O. P is the mid point of side AC and MP intersect CN in Q
To find : if the area of triangle OMQ is n then find area of triangle ABC
Solution:
Medians divides triangle into 6 Equal area triangle
=> Area of ΔOMC = Area of ΔABC/6
Median intersection point Divides Median in 2 : 1 ratio
=> OC = (2/3) CN
M & P are mid point of BC & AC
=> ΔMPC ≈ ΔABC
=> CQ/CN = 1/2
=> CQ = CN/2
OQ = OC - CQ = (2/3) CN - CN/2
=> OQ = CN/6
CQ = CN/2
OC = 2CN/3 = 4CN/6
area of Δ OMQ = (OQ/OC) area of ΔOMC
=> area of Δ OMQ = (1/4) area of ΔOMC
=> n = (1/4) area of ΔOMC
=> n = (1/4) Area of ΔABC/6
=> Area of ΔABC = 24n
Area of ΔABC = 24n
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