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Quick Check
Suppose there are some of the positions of X and Y that we have considered:
l When X is 5 mm away from S and Y is 3 cm away from R, then XY = ........ cm ........ mm.
l When X is 2 cm away from S and Y is 2 cm away from R, then XY = ........ cm ........ mm.
l When X is 3 cm away from S and Y is 5 cm away from R, then XY = ........ cm ........ mm.

Breaking Rectangles

Now we will construct a rectangle that can be divided into 2 identical squares. Let us see how this
can be done.
P X Q
First, we will draw a rough diagram of a rectangle divided into two
identical squares. Let PQRS be a rectangle divided into two identical
squares PXYS and QXYR as shown in the adjoining diagram.
Since the two squares are identical, all the shorter sides are equal, i.e.,
SP = PX = XQ = QR = RY = YS = XY. S Y R
Thus, to construct a rectangle that can be divided into two identical squares, we need to start by
selecting the shorter side of any length. Then, the longer side must be chosen so that its length is
exactly twice that of the shorter side.
The steps of the construction of such rectangle are as follows:

Step 1: Draw a horizontal line, labelled l, and mark a point S on the line.
l
S
Step 2: From point S, draw a line perpendicular Step 3: Set your compass to the length SP. Using
to l and choose a point P on this perpendicular S as the centre, draw an arc that cuts line l at a
line. The length of SP will be the side of the new point, Y. Now, using Y as the centre, draw the
square. another arc with same radius again, and mark the
point where it cuts l as R.

P P

l l
S S Y R
Step 4: From point R, draw a line perpendicular Step 5: Draw a perpendicular line from Y that cuts
to l and on this perpendicular line, mark a PQ at X. Thus, a rectangle is divided into the two
point Q such that RQ = SP. Join points P and identical squares.
Q using a ruler. The quadrilateral PQRS is
the rectangle, which can be divided into two
identical squares.

P Q P X Q

l
S Y R S Y R

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