ABCD is a trapezium ,AB is parallel to DC P and Q are the mud points of AC and - Maths - Quadrilaterals

Question

ABCD is a trapezium ,AB is parallel to DC P and Q are the mud
points of AC and BD Prove that PQ//AB//DC and PQ =1/2(AB-DC)

Lovina Kansal · Accepted Answer

Dear student
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Given: A trapezium ABCD in which AB∥DC and P and Q are the mid-pointsof its diagonals AC and BD respectively
To prove: (i) PQ∥AB or DC     (ii) PQ=12AB-DCConstruction: Join DP and produce DP to meet AB in R
Proof: Since AB∥DC and transversal AC cuts them at A and C respectively
∴∠1=∠2      
(1)     [∴Alternate angles are equal]Now, In △s APR and DPC, we have∠1=∠2      [from 1]AP=CP    ∵P is the mid-point of ACand ∠3=∠4      [Vertcally opposite angles]So, by ASA criterion of congruence, we have△APR≅△DPC⇒AR=DC and PR=DP     [C
P C T]      
(2)In △DRB,P and Q are the mid-points of sides DR and DB respectively
∴PQ∥RB⇒PA∥AB      [∵RB is a a part of AB]PQ∥AB and DC    [∵AB∥DC   (Given)]Again,P and Q are the mid-points of sides DR and DB respectively in △DRB
∴PQ=12RB⇒PQ=12AB-AR⇒PQ=12AB-DC    ∵from (2), AR=DC
Regards

ABCD is a trapezium ,AB is parallel to DC .P and Q are the mud points of AC and BD.Prove that PQ//AB//DC and PQ =1/2(AB-DC)