Concept Notes 1: Ruler & Compass Constructions: Perpendicular Bisector and Angle Bisectors
Geometric construction is the process of drawing accurate geometric figures using only two idealized Euclidean instruments:
- Unmarked Straightedge (Ruler): Used exclusively for drawing straight line segments connecting two points.
- Compass: Used exclusively for drawing circles and arcs of specified radii, and for transferring distances accurately.
1. Construction 1: Parallel Line Through an External Point
Given a line $l$ and a point $P$ not lying on $l$, to construct a line through $P$ parallel to $l$:
- Mathematical Principle: Alternate interior angles are equal when two lines are parallel ($Z$-property).
- Step-by-Step Procedure:
- Take any point $X$ on line $l$ and join $PX$.
- With $X$ as center and any convenient radius, draw an arc intersecting line $l$ at $C$ and line $PX$ at $D$.
- With $P$ as center and the same radius, draw an arc intersecting $PX$ at $E$.
- Measure distance $CD$ with compass. With $E$ as center, strike an arc of radius $CD$ to intersect the previous arc at $F$.
- Join $PF$ and extend it to form line $m$.
- Line $m \parallel l$ because alternate interior angles $\angle FPX = \angle CXP$.
2. Construction 2: Perpendicular Bisector of a Line Segment
A Perpendicular Bisector of a segment $AB$ is a line perpendicular to $AB$ passing through its exact midpoint $M$.
- Locus Property: Every point on the perpendicular bisector is equidistant from endpoints $A$ and $B$ ($PA = PB$).
- Step-by-Step Procedure:
- Given line segment $AB$.
- Set the compass opening to more than half the length of $AB$ ($\text{Radius } r > \frac{AB}{2}$).
- With center $A$, draw arcs above and below the segment.
- With center $B$ and the same radius, draw arcs intersecting the first arcs at points $P$ and $Q$.
- Draw line $PQ$.
- $PQ$ intersects $AB$ at $M$. $AM = MB$ and $\angle PMA = 90^\circ$.
[!IMPORTANT] If radius $r \le \frac{AB}{2}$, the two circles will either touch at one point or not intersect at all, making construction impossible.
3. Construction 3: Angle Bisector
The Angle Bisector divides an angle $\angle ABC$ into two congruent angles: $\angle ABD = \angle CBD = \frac{1}{2}\angle ABC$.
- Step-by-Step Procedure:
- With vertex $B$ as center and any convenient radius, draw an arc intersecting arm $BA$ at $P$ and arm $BC$ at $Q$.
- With center $P$ and radius greater than half of $PQ$, draw an arc in the interior of $\angle ABC$.
- With center $Q$ and the same radius, draw another arc intersecting the previous arc at $D$.
- Draw ray $BD$.
- Ray $BD$ is the required angle bisector.
4. Constructing Special Angles Without a Protractor
Using compass and straightedge alone, we construct exact angle benchmarks:
- $60^\circ$ Angle: Draw a ray with endpoint $O$. With center $O$, draw a large arc cutting the ray at $A$. With center $A$ and the same radius, cut the arc at $B$. Ray $OB$ forms $\angle AOB = 60^\circ$ (forms an equilateral triangle).
- $30^\circ$ Angle: Bisect the $60^\circ$ angle.
- $120^\circ$ Angle: From $B$ ($60^\circ$), swing another arc of same radius to cut at $C$. Ray $OC$ forms $\angle AOC = 120^\circ$.
- $90^\circ$ Angle (Right Angle): Bisect the angle between $60^\circ$ (point $B$) and $120^\circ$ (point $C$). Midpoint angle = $\frac{60^\circ + 120^\circ}{2} = 90^\circ$.
- $45^\circ$ Angle: Bisect the $90^\circ$ angle.
- $15^\circ$ Angle: Bisect the $30^\circ$ angle.