$ \mathrm{ABCD}$ is a trapezium in which $ \mathrm{AB} \left|\right| \mathrm{DC}, \mathrm{BD}$ is a diagonal and $ \mathrm{E}$ is the mid-point of $ \mathrm{AD}$. A line is drawn through $ \mathrm{E} $parallel to $ \mathrm{AB}$ intersecting $ \mathrm{BC}$ at $ \mathrm{F}$ (see Fig.) . Show that $ \mathrm{F} $is the mid-point of $ \mathrm{BC}.$

Open in App

Solution

Showing that $F$ is the mid-point of $BC$ :
Given that:
$ABCD$ is a trapezium in which $AB | | DC$ , $BD$ is diagonal and $E$ is the mid-point of $AD .$
Proof:
$BD$ intersected $EF$ at $G .$
In $ΔBAD,$
$E$ is the mid point of $AD$ and also $EG | | AB$ .
Thus, $G$ is the midpoint of $BD$ (Converse of midpoint theorem)
Now,
In $ΔBDC,$
$G$ is the midpoint of $BD$ and also. $GF | | AB | | DC$
Therefore, $F$ is the midpoint of $BC$ ( by Converse of midpoint theorem)
Hence proved

Suggest Corrections

Similar questions

A B C D

is a trapezium in which

A B ∥ D C, B D

is a diagonal and

E

is the mid-point of

A D

. A line is drawn through

E

parallel to

A B

, intersecting

B C

F

(see Fig). Show that

F

is the mid-point of

B C

Q. ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F ( See the given figure). Show that F is the mid - point of BC.

Q. Question 4
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F ( See the given figure). Show that F is the mid - point of BC.

$△ ABC$ is a triangle right angled at $C$ . A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$ . Show that $D$ is the mid-point of $AC$

Q. ABCD is a trapezium in which AB | | DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (Figure). Prove that F is the mid-point of BC.

Join BYJU'S Learning Program