Points E, F, and G divide the sides QP, RQ, and PR of an equilateral triangle PQR in the ratio 1:2 respectively. Find the area of triangle EFG (in square units), if the area of triangle PQR is 120 square units.
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Answers
Given : Points E, F, and G divide the sides QP, RQ, and PR of an equilateral triangle PQR in the ratio 1:2 respectively.
To Find : the area of triangle EFG (in square units), if the area of triangle PQR is 120 square units.
Solution:
Let say PQ = QR = PR = 3x
QE = x , PE = 2x
RF = x QF = 2x
PG = x GR = 2x
∠P = ∠Q = ∠R = 60°
=> ΔEQF ≅ ΔGPE ≅ ΔFRG
Lets draw EM ⊥ QR ( QF)
∠P = 60°
Sin 60° = EM/ QE
=> (√3/2) = EM/x
=> EM = x√3/2
Area of ΔEQF = (1/2) * QF * EM
= (1/2) * (2x) ( x√3/2)
= √3x²/2
ΔEQF ≅ ΔGPE ≅ ΔFRG
=> ar( ΔEQF) = ar (ΔGPE) = ar (ΔFRG ) = √3x²/2
=> (ar( ΔEQF) + ar (ΔGPE) + ar (ΔFRG ) ) = 3√3x²/2
ar (ΔEFG) = ar ( ΔPQR) - (ar( ΔEQF) + ar (ΔGPE) + ar (ΔFRG ) )
ar ( ΔPQR) = (√3 / 4) (3x)² = 9√3x²/4
9√3x²/4 = 120
=> ar (ΔEFG) = 9√3x²/4 - 3√3x²/2
=> ar (ΔEFG) = 9√3x²/4 ( 1 - 2/3)
=> ar (ΔEFG) = 9√3x²/4 ( 1/3)
=> ar (ΔEFG) = 120 ( 1/3)
=> ar (ΔEFG) = 40 sq units
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