Geometric Constructions

Cheatsheet Content

Fundamental Geometric Elements Point: A location in space with no dimensions. Represented by a dot. Line: A straight one-dimensional figure having no thickness and extending infinitely in both directions. A B Line Segment: A part of a line that is bounded by two distinct endpoints. Its length can be measured. C D Ray: A part of a line that has one endpoint and extends infinitely in one direction. E F Types of Lines Parallel Lines: Two straight lines in a plane that do not intersect at any point, maintaining a constant distance between them. L1 L2 Intersecting Lines: Two or more lines that cross each other at a common point. P Perpendicular Lines: Two lines that intersect at a $90^\circ$ angle. Concurrent Lines: Three or more straight lines that intersect at a single common point. P Types of Angles Angle: Formed by two rays sharing a common endpoint (the vertex). Right Angle: Exactly $90^\circ$. 90° Acute Angle: Less than $90^\circ$ ($0^\circ $\theta Obtuse Angle: More than $90^\circ$ but less than $180^\circ$ ($90^\circ $\theta > 90^\circ$ Straight Angle: Exactly $180^\circ$. A straight line. 180° Reflex Angle: More than $180^\circ$ but less than $360^\circ$ ($180^\circ Reflex $\theta$ Zero Angle: $0^\circ$. Two rays overlapping. Complete Angle: $360^\circ$. A full rotation. Constructing Common Angles Constructing an Angle of $60^\circ$ Draw a ray OA. With O as center and any convenient radius, draw an arc cutting OA at B. With B as center and the same radius, draw an arc cutting the first arc at C. Join O and C. $\angle AOC = 60^\circ$. O A B C 60° Constructing an Angle of $90^\circ$ Draw a ray OA. With O as center, draw a semi-circle intersecting OA at B. With B as center and the same radius, draw an arc cutting the semi-circle at C. With C as center and the same radius, draw an arc cutting the semi-circle at D. With C and D as centers and a radius greater than half of CD, draw two arcs intersecting at E. Join O and E. $\angle EOA = 90^\circ$. O A B E 90° Constructing an Angle Bisector To divide any angle into two equal parts. Draw an angle $\angle XOY$. With O as center and any convenient radius, draw an arc cutting OX at A and OY at B. With A as center and a radius greater than half of AB, draw an arc. With B as center and the same radius, draw another arc intersecting the previous arc at C. Join O and C. OC is the angle bisector of $\angle XOY$. O Y X B A C $\alpha$ $\alpha$ Constructing $45^\circ$, $30^\circ$, $120^\circ$, $150^\circ$, $75^\circ$, $105^\circ$, $135^\circ$ $45^\circ$: Bisect a $90^\circ$ angle. $30^\circ$: Bisect a $60^\circ$ angle. $120^\circ$: Construct $60^\circ$ twice on the same arc. From the initial ray, mark $60^\circ$ (point C). From point C, mark another $60^\circ$ (point D). Join O to D. $\angle AOD = 120^\circ$. O A B C D 120° $150^\circ$: Bisect the angle between $120^\circ$ and $180^\circ$. (Construct $180^\circ$ (straight line), then $120^\circ$. Bisect the angle between these two rays). $75^\circ$: Bisect the angle between $60^\circ$ and $90^\circ$. $105^\circ$: Bisect the angle between $90^\circ$ and $120^\circ$. $135^\circ$: Bisect the angle between $90^\circ$ and $180^\circ$ (on the side beyond $90^\circ$). Constructing a Perpendicular Bisector To divide a line segment into two equal parts and be perpendicular to it. Draw a line segment AB. With A as center and a radius greater than half of AB, draw arcs above and below AB. With B as center and the same radius, draw arcs above and below AB, intersecting the previous arcs at C and D respectively. Join C and D. CD is the perpendicular bisector of AB. A B C D Drawing a Perpendicular to a Given Line from a Point on the Line This construction is essentially creating a $90^\circ$ angle at a specific point on a line. Draw a line 'l' and mark a point P on it. With P as center and a small radius, draw arcs cutting line 'l' at A and B. With A and B as centers and a radius greater than AP, draw two arcs intersecting above the line 'l' at C. Join P and C. The line PC is perpendicular to 'l' at P. P A B C l Drawing a Perpendicular to a Given Line from a Point Outside the Line Draw a line 'l' and mark a point P above it. With P as center and a suitable radius, draw an arc cutting line 'l' at two points, A and B. With A and B as centers and a radius greater than half of AB, draw two arcs intersecting below line 'l' at Q. Join P and Q. The line PQ is perpendicular to 'l'. P A B Q l Constructing an Angle Equal to a Given Angle Draw a ray PQ. This will be one side of the new angle. Given angle $\angle ABC$. With B as center and a suitable radius, draw an arc cutting AB at D and BC at E. With P as center and the same radius, draw an arc cutting PQ at R. Measure the length of the arc DE with your compass. With R as center and the radius equal to arc length DE, draw an arc intersecting the previous arc at S. Join P and S. $\angle SPQ$ is equal to $\angle ABC$. B C A E D P Q S R