The sides $A B$ and $C D$ of a trapezium $A B C D$ are parallel, with $A B$ being the smaller side. $P$ is the midpoint of $CD$ and $ABPD$ is a parallelogram. If the difference between the areas of the parallelogram $ABPD$ and the triangle $BPC$ is $10$ sq cm, then the area, in sq cm, of the trapezium $A B C D$ is

We need to break down the trapezium into manageable pieces and use the special properties created by the parallelogram.

Let us start by understanding what we have:

ABCD is a trapezium with AB || CD (AB is shorter)

P is the midpoint of CD, so DP = PC

ABPD forms a parallelogram

The difference between areas of parallelogram ABPD and triangle BPC is 10 sq cm

The key insight is that this trapezium can be split into exactly two parts: the parallelogram ABPD and the triangle BPC.

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Since ABPD is a parallelogram, we can draw diagonal AP to split it into two congruent triangles:

Triangle ABP

Triangle ADP

Therefore: Area(△ABP) = Area(△ADP) = let's call this $x$

So: Area(parallelogram ABPD) = $2x$

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Here's where the magic happens! Since ABPD is a parallelogram:

AD = BP (opposite sides are equal)

AD || BP (opposite sides are parallel)

Also, since P is the midpoint of CD:

DP = PC

Now we can show that triangles ADP and BPC are congruent:

AD = BP (from parallelogram property)

DP = PC (P is midpoint)

∠ADP = ∠BPC (corresponding angles, since AD || BP and they're cut by transversal DP and PC)

By SAS congruence: △ADP ≅ △BPC

Therefore: Area(△BPC) = Area(△ADP) = $x$

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Now we have all the pieces:

Area(parallelogram ABPD) = $2x$

Area(triangle BPC) = $x$

Area(trapezium ABCD) = Area(parallelogram ABPD) + Area(triangle BPC) = $2x + x = 3x$

Given information: Area(parallelogram ABPD) - Area(triangle BPC) = 10

Substituting: $2x - x = 10$

Therefore: $x = 10$

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Area of trapezium ABCD = $3x = 3 × 10 = 30$ sq cm

When a trapezium contains a parallelogram like this, the congruent triangles created allow us to express the total area as a simple multiple of one triangle's area. This pattern appears frequently in geometry problems involving trapeziums and parallelograms.