Geometric Notation: Points, Lines and Planes

Points, Vertices and Angles

• Every time you use GPS to find a location, you are essentially pinpointing a mathematical "point" in space.
• A Point is a precise location in space with no size or thickness, represented by a single capital letter (e.g., $A$, $B$, $C$).
• A Vertex (plural: vertices) is a point where two or more line segments, edges, or curves meet.

Example Diagram: Points and Vertices

      B
     / \
    /   \
   /     \
  A-------C       x P

In the diagram above, $A$, $B$, and $C$ are the vertices of the triangle, labeled with capital letters. $P$ is a separate point in space, marked with a cross and a capital letter.

• In AQA exam papers, the vertices of a polygon are labeled in consecutive order around the perimeter, typically anti-clockwise (e.g., a square labeled $ABCD$).
• In 3D coordinate geometry, a vertex is written as $(x, y, z)$.
• If multiple points lie on the exact same straight line, they are called Collinear Points.
• Angles are named using three capital letters where the middle letter represents the vertex (e.g., $\angle ABC$), or by a single capital letter with a hat denoting the vertex (e.g., $\hat{B}$).

Lines, Line Segments and Edges

• You can snap a piece of spaghetti to make a finite segment, but a mathematical line goes on forever.
• A continuous Line extends infinitely in both directions and is labeled with a single lower-case letter (e.g., line $l$).
• A Line Segment is a finite part of a line bounded by two distinct endpoints, named by its two capital letters (e.g., $AB$).
• The notation $AB$ can refer to the object itself, while $AB = 5\text{ cm}$ describes its measured length.
• An Edge is a specific line segment where two faces of a 3D solid meet, or the boundary segment of a 2D polygon (e.g., a cube has exactly 12 edges).

Parallel Lines and Transversals

• Think of train tracks that never cross compared to a crossroads where two streets meet.
• Parallel Lines are straight lines in the same plane that never intersect and remain equidistant at all points.
• They are denoted by the symbol $\parallel$ (e.g., $AB \parallel CD$) and are indicated on diagrams by matching arrowheads (e.g., $>$ or $>>$).
• A Transversal is a line that crosses at least two other lines.
• When a transversal crosses parallel lines, specific angle rules apply: alternate angles are equal, corresponding angles are equal, and allied (co-interior) angles sum to $180^\circ$.
• For higher tier coordinate geometry, two lines are parallel if they have the exact same gradient ($m_1 = m_2$).

Perpendicular Lines

• Perpendicular Lines are two lines that meet or intersect exactly at a $90^\circ$ angle.
• They are denoted by the symbol $\perp$ (e.g., $AB \perp BC$) and are marked on diagrams with a small square at the intersection.
• A Normal is a specific line perpendicular to a tangent at the point of contact on a curve or circle.
• A Perpendicular Bisector is a line that divides a segment into two equal halves at a strict $90^\circ$ angle.

Worked Example: Perpendicular Gradients

In coordinate geometry, the product of the gradients of two perpendicular lines must always be $-1$. The perpendicular gradient is the negative reciprocal of the original gradient.

Prove that Line 1 ($y = 0.4x - 3$) and Line 2 ($5x + 2y = 10$) are perpendicular.

Step 1: State the formula for perpendicular gradients.

• $m_1 \times m_2 = -1$

Step 2: Identify the gradient ($m$) for both lines.

• Line 1 is already in $y = mx + c$ format: $m_1 = 0.4$
• Line 2 must be rearranged: $2y = -5x + 10 \Rightarrow y = -2.5x + 5$. Therefore, $m_2 = -2.5$

Step 3: Substitute the gradients into the formula.

• $0.4 \times -2.5 = -1$

Step 4: State your conclusion clearly.

• Since the product of the gradients is exactly $-1$, the lines are perpendicular.

Planes in 3D Geometry

• A sheet of paper is a physical model for a plane, except a true mathematical plane has zero thickness and extends indefinitely.
• A Plane is a flat, 2D surface. Points that lie on the exact same plane are referred to as Coplanar.
• A Face is the finite portion of a plane forming the flat surface of a 3D solid.
• A Plane of Symmetry is a 2D plane that divides a 3D shape into two identical mirror images (e.g., a cube has 9 planes of symmetry).
• Planes are typically named using the capital letters of their vertices in perimeter order (e.g., plane $ABCD$) or a single capital letter in the corner (e.g., plane $P$).

Congruence and Symmetry Notation

• How do you know a shape is an isosceles triangle just by looking at a sketch? Look for the secret code of dashes and arcs.
• Short straight dashes drawn through line segments indicate they are of equal length (e.g., a single dash $|$ on two sides means they are exactly equal in length).
• Arcs drawn inside the vertices of a shape indicate equal angles.
• Congruent shapes are exactly the same shape and size, meaning all corresponding sides and angles are equal. This is denoted by the symbol $\cong$.
• A Regular Polygon is a polygon where all sides are equal (marked with dashes) and all interior angles are equal (marked with arcs).
• A Line of Symmetry divides a 2D shape into two congruent mirror images, often indicated by a dashed or dotted line.
• The Order of Rotational Symmetry is the number of positions in which a shape looks identical to its original position during a full $360^\circ$ rotation.