Mathematics: Class 6 — Chapter 8: Playing with Constructions

A curve is any shape that can be drawn on paper with a pencil, including straight lines, circles, and other freeform figures.

To draw a circle of a specific radius (e.g., $4\text{ cm}$) around a point $P$:

• Mark a point $P$ as the centre.
• Use a ruler to open the compass to a distance of $4\text{ cm}$ between the tip and the pencil.
• Keep the tip fixed at $P$ and rotate the pencil to draw the full curve.
• The distance from the centre $P$ to any point on the circle is the radius.

• A Person: A figure composed of a circle (head) and a body with curved lines. The challenge lies in finding where to place the compass tip and what radius to use for the curved neck.
• Wavy Wave: Constructing waves on a central line $AB = 8\text{ cm}$ using half circles. For a wave of length $8\text{ cm}$, the radius of each half-circle wave is $2\text{ cm}$, and the segment length $AX = 4\text{ cm}$.
• Eyes: Constructing symmetrical eye shapes using intersecting arcs.

For a rectangle (e.g., $ABCD$):

• R1: The opposite sides are equal in length ($AB = CD$ and $AD = BC$).
• R2: All angles are right angles ($90^\circ$).

For a square:

• S1: All four sides are equal in length ($AB = BC = CD = DA$).
• S2: All angles are right angles ($90^\circ$).

In a valid name for a square or rectangle, the corners must be listed in the order of travel around the boundary, starting from any corner (e.g., $ABCD$, $BCDA$, $ADCB$). Non-consecutive naming like $ABDC$ or $ACBD$ is invalid.

Rotating a square or rectangle does not change its side lengths or angles. A rotated square is still a square, and a rotated rectangle is still a rectangle.

Consider a rectangle $ABCD$ with $AB = 7\text{ cm}$ and $BC = 4\text{ cm}$. Let $X$ be a point on $AD$ and $Y$ be a point on $BC$. By varying the positions of $X$ and $Y$, we can measure the changing length of $XY$:

• When $X$ is at $A$ and $Y$ is at $B$, $XY = AB = 7\text{ cm}$ (minimum distance).
• When $X$ and $Y$ are at equal distances from $A$ and $B$ respectively, $ABYX$ forms a rectangle, and $XY = AB = 7\text{ cm}$.
• The maximum distance between $X$ and $Y$ occurs when they are at opposite corners (diagonals $AC$ or $BD$).

Constructing a rectangle that can be divided into identical squares:

• To divide a rectangle into $2$ identical squares of side $a$, the dimensions of the rectangle must be $a \times 2a$.
• To divide a rectangle into $3$ identical squares of side $a$, the dimensions must be $a \times 3a$.
• A compass can be used to transfer the length of one side multiple times along a perpendicular line to mark out identical squares without a ruler.

• The diagonals of a rectangle are equal in length ($PR = QS$).
• A diagonal divides each opposite angle into two smaller angles (e.g., $\angle P$ is divided into $c$ and $d$; $\angle R$ is divided into $g$ and $h$). In a general rectangle, $c \neq d$ and $g \neq h$.
• In a square, the diagonal divides the opposite angles into equal parts ($45^\circ$ each).

Example 1: Construct a rectangle where a diagonal divides the opposite angles into $60^\circ$ and $30^\circ$:

• Draw base $AB$ of arbitrary length.
• Draw a perpendicular at $B$.
• Draw a line from $A$ at an angle of $60^\circ$ to $AB$. The intersection of this line with the perpendicular from $B$ gives point $C$.
• Draw a perpendicular to $BC$ at $C$ (or use a compass with radius $AD = BC$ from $A$) to locate point $D$. Join $CD$ to complete the rectangle $ABCD$.

Example 2: Construct a rectangle with one side $5\text{ cm}$ and diagonal $7\text{ cm}$:

• Draw base $CD = 5\text{ cm}$.
• Draw a perpendicular line $l$ to $CD$ at point $C$.
• Set the compass to a radius of $7\text{ cm}$. With $D$ as the centre, draw an arc intersecting line $l$ at point $B$.
• Construct perpendiculars to $DC$ and $BC$ through $D$ and $B$ respectively to find the intersection point $A$, completing rectangle $ABCD$.

To construct a house shape where all border lines are of length $5\text{ cm}$:

• Step 1: Draw the base $DE = 5\text{ cm}$, and perpendicular sides $DB = 5\text{ cm}$ and $EC = 5\text{ cm}$.
• Step 2: To locate the roof peak $A$ which must be exactly $5\text{ cm}$ from both $B$ and $C$:
• Set the compass to a radius of $5\text{ cm}$.
• With $B$ as the centre, draw an arc.
• With $C$ as the centre, draw another arc.
• The intersection of these two arcs is the point $A$.
• Step 3: Join $AB$ and $AC$ with straight lines.
• Step 4: Draw the curved ceiling arc touching $B$ and $C$ by placing the compass tip at $A$ with a radius of $5\text{ cm}$.