Geometry and Measurement - Geometric Constructions

Introduction to Geometric Construction: Construction is the process of drawing geometric figures accurately using only two tools: a compass and a straightedge (ruler without markings). In these drawings, we do not use a protractor to measure angles, but rather use geometric properties to create them. A straightedge is used to draw line segments, while a compass is used to draw circles and arcs of a specific radius.

Perpendicular Bisector: This construction produces a line that divides a segment $AB$ into two equal halves at a $90^{\circ}$ angle. To visualize the process: place the compass point at $A$ with a radius greater than half the length of $AB$ and draw arcs above and below the segment. Repeat this from point $B$ with the same radius. Connecting the two points where the arcs intersect creates the perpendicular bisector.

Angle Bisector: An angle bisector is a ray that divides an angle into two equal parts. Visually, you place the compass on the vertex $V$ and draw an arc cutting both arms of the angle at points $X$ and $Y$. Then, from $X$ and $Y$, draw two arcs with the same radius that intersect at a point $Z$ inside the angle. The line $VZ$ is the bisector, meaning $\angle XVZ = \angle YVZ$.

Constructing $60^{\circ}$ and $120^{\circ}$ Angles: These are fundamental angles built using equilateral triangle properties. Draw a ray with endpoint $O$ and an arc crossing it at $P$. Keep the compass width the same, place the point at $P$, and draw another arc crossing the first at $Q$. The angle $\angle POQ$ is $60^{\circ}$. If you move the compass to $Q$ and draw a third arc crossing the original large arc at $R$, the angle $\angle POR$ is $120^{\circ}$.

Constructing $90^{\circ}$ and $45^{\circ}$ Angles: A $90^{\circ}$ angle is typically constructed by creating a perpendicular bisector on a straight line or by bisecting the $180^{\circ}$ angle of a straight line. To create a $45^{\circ}$ angle, you first construct a $90^{\circ}$ angle and then apply the angle bisector method to split it exactly in half.

Constructing Parallel Lines: To draw a line through point $P$ parallel to line $m$, we use the property of corresponding angles. Draw a transversal line through $P$ that intersects $m$ at point $A$. Construct an arc at point $A$ and a matching arc at point $P$. Measure the width of the angle at $A$ with the compass and transfer that width to the arc at $P$. Drawing a line through $P$ and this new intersection point results in parallel lines.

Perpendicular from a Point to a Line: To drop a perpendicular from point $P$ (not on line $L$) to line $L$, draw an arc from $P$ that cuts line $L$ at two points, $X$ and $Y$. Then, find the perpendicular bisector of the segment $XY$. The line passing through $P$ and the midpoint of $XY$ will be perpendicular to line $L$ at a $90^{\circ}$ angle.