Practical Geometry - Constructing Perpendiculars

Definition of Perpendicularity: Two lines are said to be perpendicular if they intersect at a right angle of $90^{\circ}$. Visually, this forms a perfect 'L' shape or a cross '+' where all four angles are equal. If line $l$ is perpendicular to line $m$, we write it as $l \perp m$.

Perpendicular through a Point on the Line (Compasses): To construct a perpendicular at a point $P$ lying on line $l$, place the compass pointer at $P$ and draw an arc cutting $l$ at two points, $A$ and $B$. Then, from $A$ and $B$, draw two arcs with a radius greater than $AP$ that intersect at point $Q$. Connecting $P$ and $Q$ results in a vertical line $PQ$ standing exactly at $90^{\circ}$ to the horizontal line $l$.

Perpendicular through a Point outside the Line (Compasses): Given a point $P$ not on line $l$, draw an arc with $P$ as the center that cuts line $l$ at two points, $X$ and $Y$. From $X$ and $Y$, using the same radius, draw arcs on the opposite side of the line to intersect at point $Q$. The line segment $PQ$ is the perpendicular dropped from $P$ to line $l$.

Perpendicular through a Point (Set-Squares): This method uses the right-angled triangular tool from the geometry box. Place a ruler along the line $l$ and align one edge of the set-square with the ruler. Slide the set-square until its vertical edge touches point $P$. Drawing a line along this vertical edge creates a $90^{\circ}$ intersection.

The Perpendicular Bisector: A perpendicular bisector is a line that divides a given line segment into two equal halves at a $90^{\circ}$ angle. Visually, if you have a segment $AB$, the bisector passes through the exact midpoint $M$ such that $AM = MB$. It looks like a symmetrical cross where the vertical arm splits the horizontal arm into two identical pieces.

Constructing a Perpendicular Bisector: For a segment $AB$, set the compass to a radius slightly more than half the length of $AB$. Draw arcs above and below the segment from point $A$, then repeat the same from point $B$ without changing the radius. The points where these arcs intersect (above and below the line) are joined to form the perpendicular bisector.

Properties of the Perpendicular Bisector: Every point on the perpendicular bisector is at an equal distance from the endpoints of the line segment. If point $P$ lies on the bisector of $AB$, then the distance $PA = PB$.