In the given figure, line segment BE intersect the side CD of a triang

1.	In the given figure, line segment BE intersect the side CD of a triangle ACD at F. If DF: CF = 1:2, then whichof the following is always true for AB/BD? i)3AE/2CE ii)2AE/CE iii)3AE/CE iv)2AE/3CE
Answer» Answer: Option (ii) is correct AB / BD = 2 AE / CE Explanation: Given : DF : CF = 1 : 2 Construction : Refer to the attachment for figure Draw a line DG parallel to FE such that G lie on line AE Solution : CONSIDER Δ DCG since, FE ║ DG therefore, By Basic proportionality theorem → ( GE/CE ) = ( DF/CF ) → ( GE/CE ) = 1/2 → GE = CE / 2 ______equation (1) Consider Δ ABE since, DG ║ BE therefore, By basic proportionality theorem → ( AB/BD ) = ( AE/GE ) USING equation (1) to substitute the value of GE → ( AB/BD ) = ( AE / ( CE/2 ) ) → ( AB/BD ) = ( AE × 2 / CE ) → ( AB/BD ) = ( 2 AE / CE ) therefore, Option (ii) 2 AE/CE is correct . STATEMENT of Basic Proportionality theorem if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the other two sides are divided in the same ratio.

In the given figure, line segment BE intersect the side CD of a triangle ACD at F. If DF: CF = 1:2, then whichof the following is always true for AB/BD? i)3AE/2CE ii)2AE/CE iii)3AE/CE iv)2AE/3CE

Answer»

AB / BD = 2 AE / CE

Given :

Construction :

Refer to the attachment for figure

Solution :

CONSIDER Δ DCG

since, FE ║ DG therefore, By Basic proportionality theorem

→ ( GE/CE ) = ( DF/CF )

→ ( GE/CE ) = 1/2

→ GE = CE / 2 ______equation (1)

Consider Δ ABE

since, DG ║ BE therefore, By basic proportionality theorem

→ ( AB/BD ) = ( AE/GE )

USING equation (1) to substitute the value of GE

→ ( AB/BD ) = ( AE / ( CE/2 ) )

→ ( AB/BD ) = ( AE × 2 / CE )

→ ( AB/BD ) = ( 2 AE / CE )

therefore,

if a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the other two sides are divided in the same ratio.