Key Points

Standard geometric constructions are performed using only an unmarked ruler (for drawing straight lines) and a compass (for drawing arcs, circles, and transferring lengths).

Any point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the segment. If point P is on the perpendicular bisector of segment XY, then the length $PX = PY$.

To construct the perpendicular bisector of a line segment XY, draw arcs with the same radius (more than half the length of XY) from both X and Y on both sides of the line. The line joining the two intersection points of the arcs is the perpendicular bisector.

To construct a $90^\circ$ angle at a point O on a line, first mark two points X and Y on the line that are equidistant from O. Then, construct the perpendicular bisector of the segment XY, which will pass through O and form a right angle.

To bisect an angle, draw an arc from the vertex that intersects both arms of the angle. From these two intersection points, draw two more arcs of equal radius that intersect each other. A line from the vertex to this new intersection point bisects the angle.

A $60^\circ$ angle is constructed by creating an equilateral triangle. Draw a ray, then from its endpoint, draw an arc. From the point where the arc intersects the ray, draw another arc of the same radius to intersect the first one. The angle formed is $60^\circ$.

Many standard angles are constructed by combining or bisecting other known angles. For example, a $45^\circ$ angle is made by bisecting a $90^\circ$ angle. A $30^\circ$ angle is made by bisecting a $60^\circ$ angle.

An angle can be copied using a compass and ruler by constructing a congruent triangle. The method relies on the SSS (Side-Side-Side) congruence condition to ensure the new angle is identical to the original.

To construct a line parallel to a given line through an external point, first draw a transversal line. Then, copy either a corresponding angle or an alternate interior angle at the external point. The new line will be parallel to the original.

A regular hexagon has six equal sides and six equal interior angles, each measuring $120^\circ$. It can be divided into six congruent equilateral triangles meeting at the center.

To construct a regular hexagon, draw a circle and, without changing the compass radius, mark six consecutive points on the circumference. Joining these points creates a regular hexagon with a side length equal to the radius of the circle.

Tiling, also called tessellation, is the process of covering a surface with one or more geometric shapes (tiles) without any gaps or overlaps.

An $m \times n$ rectangular grid can be tiled by $2 \times 1$ tiles (dominoes) if and only if the total area, $m \times n$, is an even number. This means at least one of the dimensions, $m$ or $n$, must be even.

A powerful method to prove that a tiling is impossible is to color the grid like a chessboard. Since a $2 \times 1$ tile always covers one white and one black square, a region cannot be tiled if it has an unequal number of black and white squares.

Only three types of regular polygons can tile the entire plane by themselves: equilateral triangles ($60^\circ$ angles), squares ($90^\circ$ angles), and regular hexagons ($120^\circ$ angles). This is because the angles at any vertex where tiles meet must sum to exactly $360^\circ$.