Question

In the given figure, line segment BE intersect the side CD of a triangle ACD at F. If DF: CF = 1:2, then which

of the following is always true for AB/BD? i)3AE/2CE ii)2AE/CE iii)3AE/CE iv)2AE/3CE​

Answer 1

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.