Geometry - Practical Geometry: Construction using Ruler and Compasses

Construction of a Circle: To construct a circle of a given radius $r$, mark a center point $O$. Place the metal pointer of the compass at $O$ and open the compass legs to the measure of $r$ using a ruler. Keeping the pointer fixed, rotate the pencil arm $360^\circ$ to form a closed curve where every point is at distance $r$ from $O$.

Line Segment of Given Length: To draw a line segment $AB$ of length $l$, draw a long line $m$ and mark a point $A$. Adjust the compass to the length $l$ on a ruler. Place the pointer at $A$ and draw an arc that intersects line $m$ at point $B$. The distance between $A$ and $B$ is exactly $l$.

Perpendicular Bisector of a Line Segment: This is a line that divides a segment $AB$ into two equal halves at a $90^\circ$ angle. Visually, this is achieved by drawing two arcs with a radius greater than $\frac{1}{2} AB$ centered at $A$ and $B$. These arcs intersect at two points, one above and one below the segment. Connecting these intersection points creates the perpendicular bisector.

Perpendicular to a Line through a Point: To draw a perpendicular to line $L$ from a point $P$ not on it, draw an arc from $P$ that cuts line $L$ at two points $X$ and $Y$. From $X$ and $Y$, draw two intersecting arcs on the opposite side of the line. The line joining $P$ and this intersection is perpendicular to $L$ at $90^\circ$.

Construction of Standard Angles: Angles like $60^\circ$ are constructed by drawing an initial arc from a vertex $O$ cutting the base line at $A$. Without changing the compass width, draw another arc from $A$ to intersect the first arc at $B$. The angle $\angle AOB$ is exactly $60^\circ$. By repeating this, one can construct $120^\circ$.

Angle Bisector: To divide an angle $\angle ABC$ into two equal parts, draw an arc centered at $B$ that intersects arms $BA$ and $BC$ at points $P$ and $Q$ respectively. From $P$ and $Q$, draw two arcs with the same radius that intersect at a point $R$ in the interior of the angle. The ray $BR$ is the angle bisector, such that $\angle ABR = \angle CBR$.

Construction of $90^\circ$ and $45^\circ$: A $90^\circ$ angle is constructed by bisecting the $180^\circ$ straight angle or by bisecting the angle between $60^\circ$ and $120^\circ$ arcs. To obtain $45^\circ$, construct a $90^\circ$ angle and then bisect it using the angle bisector method.