Playing with Constructions - Squares and Rectangles

Construct a square $ABCD$ where each side is $4\text{ cm}$.

1. Draw a line segment $AB = 4\text{ cm}$.
2. At point $A$, construct a ray $AX$ perpendicular to $AB$ using a compass ($90^\circ$ angle).
3. Use the compass to measure $4\text{ cm}$ and draw an arc from point $A$ on ray $AX$ to mark point $D$.
4. From point $B$, draw an arc of radius $4\text{ cm}$.
5. From point $D$, draw an arc of radius $4\text{ cm}$ to intersect the previous arc at point $C$.
6. Join $BC$ and $CD$.

Since all sides of a square are equal ($4\text{ cm}$) and all angles are $90^\circ$, we ensure the base and height are perpendicular and all segments are exactly $4\text{ cm}$ long.

Construct a rectangle $PQRS$ with length $PQ = 6\text{ cm}$ and breadth $QR = 3\text{ cm}$.

1. Draw a line segment $PQ = 6\text{ cm}$.
2. At $Q$, construct a perpendicular ray $QY$.
3. With $Q$ as center and radius $3\text{ cm}$, draw an arc cutting $QY$ at point $R$.
4. With $R$ as center and radius $6\text{ cm}$, draw an arc.
5. With $P$ as center and radius $3\text{ cm}$, draw an arc to intersect the previous arc at point $S$.
6. Join $RS$ and $PS$.

In a rectangle, opposite sides are equal. Thus $PQ = RS = 6\text{ cm}$ and $QR = PS = 3\text{ cm}$. The construction uses the $90^\circ$ property to ensure the shape is rectangular.

1. Draw a line segment $WX = 5\text{ cm}$.
2. At point $W$, construct a ray $WP$ perpendicular to $WX$ (making $90^\circ$).
3. From point $W$, cut an arc of $5\text{ cm}$ on ray $WP$ to mark point $Z$.
4. At point $X$, construct another ray $XQ$ perpendicular to $WX$.
5. From point $X$, cut an arc of $5\text{ cm}$ on ray $XQ$ to mark point $Y$.
6. Join $ZY$ to complete the square $WXYZ$.

Since all sides of a square are equal and all angles are $90^\circ$, we use the side length $5\text{ cm}$ to define all four vertices after establishing right angles at the base.

1. Draw a line segment $KL = 7\text{ cm}$.
2. At point $L$, construct a $90^\circ$ angle and draw a ray $LR$.
3. On ray $LR$, mark a point $M$ such that $LM = 4\text{ cm}$.
4. From point $M$, draw an arc of $7\text{ cm}$ towards the left.
5. From point $K$, draw an arc of $4\text{ cm}$ upwards to intersect the previous arc at point $N$.
6. Join $MN$ and $NK$. $KLMN$ is the required rectangle.

Opposite sides of a rectangle are equal ($KL=MN=7\text{ cm}$ and $LM=NK=4\text{ cm}$) and all angles are $90^\circ$. This construction uses both side measurements and the perpendicular property.