Geometry Terms and Notation for the TMUA

Fundamental Elements: Points, Lines, and Planes

In geometric problems, it is vital to distinguish between different types of linear elements. A point represents a specific position in space and has no dimensions. When we connect two points, we form a line segment, which has a finite length and two distinct endpoints. In contrast, a line is understood to extend infinitely in both directions. In three dimensional geometry, we encounter planes, which are flat, two dimensional surfaces that extend infinitely. When dealing with solid shapes, we refer to the corners as vertices (singular: vertex) and the lines where two faces meet as edges.

Relationships Between Lines

Lines and line segments can relate to each other in specific ways that define the geometry of a shape. Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. In coordinate geometry, parallel lines share the same gradient. Perpendicular lines are lines that intersect at a right angle, which is exactly $90^{\circ}$. When a line is perpendicular to another, the product of their gradients is $-1$ (excluding the case of vertical and horizontal lines).

Angles and Subtended Angles

An angle is formed by the intersection of two lines or segments. A critical term often used in TMUA geometry is the subtended angle. To say an angle is subtended by an arc or a line segment is to describe the angle formed when lines are drawn from the endpoints of that segment to a common point. For example, in circle geometry, a chord $AB$ subtends an angle $\angle APB$ at a point $P$ on the circumference. The magnitude of this angle depends on the position of $P$ and the length of the chord $AB$.

Polygons and Regular Polygons

A polygon is a closed, two dimensional shape with at least three straight sides. Common polygons include triangles, quadrilaterals, pentagons, and hexagons. A regular polygon is a specific type of polygon where all sides are of equal length and all interior angles are equal. For a regular polygon with $n$ sides, the sum of the interior angles is given by $(n - 2) \times 180^{\circ}$. Therefore, each individual interior angle in a regular $n$-gon is $\frac{(n - 2) \times 180^{\circ}}{n}$. Similarly, the sum of the exterior angles of any convex polygon is always $360^{\circ}$, meaning each exterior angle of a regular polygon is $\frac{360^{\circ}}{n}$.

Reflection and Rotational Symmetries

Symmetry describes how a shape remains unchanged under certain transformations. Reflection symmetry (or line symmetry) occurs if a line can be drawn through the polygon such that one half is a mirror image of the other. Rotational symmetry occurs if the polygon can be rotated around its centre by an angle less than $360^{\circ}$ and still look exactly the same. The order of rotational symmetry is the number of times the shape looks the same during a full $360^{\circ}$ turn. For any regular polygon with $n$ sides, there are exactly $n$ lines of reflection symmetry and the order of rotational symmetry is $n$. For example, a square (a regular quadrilateral) has four lines of symmetry and rotational symmetry of order four.