You visited us 1 times! Enjoying our articles? Unlock Full Access!

A Parallelogram and a Triangle between the Same Parallels

Question 8In ...

Question

Question 8
In trapezium ABCD, AB || DC and L is the midpoint of BC. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. prove that ar (ABCD) = ar (APQD).

Open in App

Solution

Given,
In trapezium ABCD, AB || DC, DC produced to Q and L is the midpoint of BC.
$∴$ BL = CL
Proof:
DC produced to Q and AB || DC.
So, DQ || AB.
In $Δ C L Q$ and $Δ B L P$ ,
CL = BL [L is the mid point of BC]
$∠ L C Q = ∠ L B P$ [alternate interior angles as BC is a transversal]
$∠ C Q L = ∠ L P B$ [alternate interior angles as PQ is a transversal]
$Δ C L Q ≅ Δ B L P$ [by AAS congruence rule]
Then, $a r (Δ C L Q) = a r (Δ B L P)$ …(i)
Now, $a r (A B C D) = a r (A P Q D) - a r (Δ C Q L) + a r (Δ B L P)$
$= a r (A P Q D) - a r (Δ B L P) + a r (Δ B L P)$ [from Eq. (i)]
$\Rightarrow$ ar (ABCD) = ar (APQD)

Suggest Corrections

Similar questions

Q. Question 8
In trapezium ABCD, AB || DC and L is the midpoint of BC. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. prove that ar (ABCD) = ar (APQD).

In a trapezium ABCD,AB||DC and M is the midpoint of BC.Through M,a line PQ||AD has been drawn which meets AB in P and DC produced in Q,as shown in the adjoining figure. Prove that ar(ABCD)=ar(APQD).

Q. In a trapezium

A B C D,

A B ∥ D C

and

L

in the mid-point of

B C .

Through

L,

a line

L M

is drawn parallel to

A D

which meets

A B

X

and

D C

produced in

Y .

prove that

a r (A B C D) = a r (A X Y D)

Q. In trapezium ABCD, AB || DC and L is the midpoint of BD. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. Prove that ar(trap. ABCD) = ar(||gm APQD).

Q. In the figure,

□ A B C D

is a trapezium,

A B ∥ D C

. Point

P

and

Q

are midpoints of

s e g A D

and

s e g B C

respectively. Then prove that,

P Q ∥ A B

and

P Q = \frac{1}{2} (A B + D C)

Join BYJU'S Learning Program

Question 8 In trapezium ABCD, AB || DC and L is the midpoint of BC. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. prove that ar (ABCD) = ar (APQD).

Question 8
In trapezium ABCD, AB || DC and L is the midpoint of BC. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. prove that ar (ABCD) = ar (APQD).