How do we find the area of a parallelogram by dissecting it into a rectangle?
Answer
The dissection turns parallelogram ABCD into rectangle ABYX, so their areas are equal — and a rectangle's area we already know.
Area of parallelogram ABCD = Area of rectangle ABYX
= AX × XY
where AX is the height of the parallelogram
Once XY is shown to equal DC (see the next question), this becomes
Area of a parallelogram = base × height
Q2.
Is there a relation between XY and DC?
Answer
Yes: XY = DC.
From the dissection, DX = CY (they are matching sides of the congruent triangles ∆AXD and ∆BYC)
Add the common part XC to both:
DX + XC = CY + XC DC = XY
Why it happens: the triangle that leaves the left end takes the piece DX with it and puts it down as CY at the right end. So the base of the rectangle is the base of the parallelogram slid along — the same length, only shifted. That is why the base of the parallelogram may be used directly in the formula.
Q3.
Can the area of the parallelogram be determined by taking another side as the base and its corresponding height?
Answer
Yes. Any side may be used as the base, provided its own height is used with it.
Using DC as base with height AX: Area = DC × AX
Using AD as base with height CZ: Area = AD × CZ
Both give the same area, so DC × AX = AD × CZ
Why it happens: the parallelogram has only one area, so every correct base–height pair must give the same number. A long side goes with a short height and a short side goes with a long height; the product is fixed. This is exactly the relation used to find an unknown height from a known one.
Q4.
Can the parallelogram be cut along CZ and rearranged to form a rectangle?
Answer
Yes. Take AD as the base and let CZ be the perpendicular from C to AD (extended if necessary).
Cutting along CZ and sliding the piece across gives a rectangle with base AD and height CZ, by exactly the same congruence argument as before — the triangle cut off at one end is congruent to the gap at the other.
Area of the parallelogram = AD × CZ
So the choice of side is free. Whichever side you can measure, together with the height to it, will do.