Geometry for ESAT Mathematics 1

Conventional Terms and Notation

In ESAT Mathematics 1, geometry is defined using precise terms. A point represents a singular position. A line is an infinite one dimensional figure, while a line segment is a finite portion of that line. Vertices are the corners where edges meet. In flat shapes, these are intersections of sides, while in figures like pyramids, the apex is also a vertex.

An edge is the side of a polygon or polyhedron. A plane is a flat surface. Lines that remain a constant perpendicular distance apart are parallel, indicated by arrows. Lines meeting at $90^\circ$ are perpendicular.

Polygons are closed plane shapes with three or more straight sides. A regular polygon has all sides and angles equal. Reflection symmetry occurs if a shape can be folded exactly onto itself. Rotational symmetry is the number of times a shape fits into its outline during a $360^\circ$ turn.

For example, a regular hexagon has 6 lines of symmetry and an order of rotational symmetry of 6. A regular octagon, similarly, has 8 lines of symmetry and an order of 8.

Angle Properties

Angle calculation is a core skill. The following rules must be mastered:

1. Angles around a point sum to $360^\circ$.
2. Angles on a straight line sum to $180^\circ$.
3. Vertically opposite angles at a vertex are equal ($a = c, b = d$).

Parallel Lines

When a transversal crosses parallel lines:

• Alternate angles are equal (Z shape).
• Corresponding angles are equal (F shape).
• Allied or co-interior angles sum to $180^\circ$ (C shape).

Polygons

The sum of interior angles in an $n$ sided polygon is $180(n - 2)$. The sum of exterior angles for any polygon is $360^\circ$. For a triangle, the sum of interior angles is $180^\circ$, and an exterior angle equals the sum of the two opposite interior angles. Quadrilaterals sum to $360^\circ$.

Example: Finding an unknown angle in a pentagon. Three angles are $100^\circ$, $110^\circ$, and $82^\circ$. The other two are equal. Sum = $180(5 - 2) = 540^\circ$. Sum of known angles = $100 + 110 + 82 = 292^\circ$. Remaining = $540 - 292 = 248^\circ$. Each equal angle = $248 / 2 = 124^\circ$.

Types of Triangles and Quadrilaterals

Shapes are identified by their side and angle properties.

• Square: 4 equal sides, 4 right angles, 2 pairs of parallel sides.
• Rectangle: 2 pairs of parallel sides, 4 right angles.
• Parallelogram: 2 pairs of parallel sides, opposite sides and angles equal.
• Trapezium: Exactly one pair of parallel sides.
• Kite: 2 pairs of equal adjacent sides, diagonals meet at right angles.
• Rhombus: 4 equal sides, opposite sides parallel, diagonals bisect at right angles.

Triangles include Acute (all angles $< 90^\circ$), Right angled (one $90^\circ$ angle), and Obtuse (one angle $> 90^\circ$). Isosceles triangles have 2 equal sides and angles, while Equilateral triangles have 3 equal sides and angles ($60^\circ$).

Congruence and Similarity

Congruence

Two triangles are congruent if they are identical in size and shape. There are four criteria:

1. SSS (Side, Side, Side): All three sides are equal.
2. SAS (Side, Angle, Side): Two sides and the included angle are equal.
3. ASA (Angle, Side, Angle): Two angles and a corresponding side are equal.
4. RHS (Right angle, Hypotenuse, Side): A right angle, the hypotenuse, and one other side are equal.

Similarity

Similar figures have the same shape but different sizes. Angles remain identical, but lengths are in a constant ratio $n$. If the length scale factor is $n$, then the area scale factor is $n^2$ and the volume scale factor is $n^3$.

Example: Scaling a chocolate bar. An individual bar has a mass of $25$ g and a wrapper area of $60$ cm$^2$. A family bar has dimensions twice as large. Scale factor $n = 2$. Area of wrapper = $60 \times 2^2 = 240$ cm$^2$. Mass (volume proportional) = $25 \times 2^3 = 200$ g.

Transformations

Transformations describe the movement of an object to an image.

• Rotation: Defined by a centre, angle, and direction (anticlockwise is positive).
• Reflection: Defined by a mirror line. Points on the line are invariant.
• Translation: Defined by a vector $\binom{a}{b}$.
• Enlargement: Defined by a centre and scale factor. Negative factors place the image on the opposite side of the centre.

Pythagoras Theorem and Circle Geometry

Pythagoras Theorem states $a^2 + b^2 = c^2$ for right angled triangles. In 3D, the diagonal $d$ of a cuboid is $d^2 = a^2 + b^2 + c^2$.

Circle Theorems

Key theorems for the ESAT include:

1. The angle at the centre is twice the angle at the circumference.
2. The angle in a semicircle is $90^\circ$.
3. Angles in the same segment are equal.
4. Opposite angles in a cyclic quadrilateral sum to $180^\circ$.
5. The angle between a tangent and a radius is $90^\circ$.
6. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Coordinate Geometry, Bearings, and Maps

On a coordinate grid, the distance between $(a, b)$ and $(c, d)$ is $\sqrt{(c-a)^2 + (d-b)^2}$. The midpoint is $(\frac{a+c}{2}, \frac{b+d}{2})$.

Bearings are measured clockwise from North and written with three digits (e.g., $070^\circ$).

Scale Drawings use ratios. For a map scale of $1:25000$, $1$ cm on the map represents $25000$ cm ($250$ m) in reality.

Mensuration: Area and Volume

• Triangle Area: $\frac{1}{2} \times \text{base} \times \text{height}$.
• Trapezium Area: $\frac{1}{2}h(a+b)$.
• Circle Area: $\pi r^2$; Circumference: $2\pi r$.
• Sector Area: $\frac{x}{360} \pi r^2$; Arc Length: $\frac{x}{360} 2\pi r$.
• Prism Volume: $\text{Area of cross-section} \times \text{length}$.
• Cylinder Volume: $\pi r^2 h$.
• Cone Volume: $\frac{1}{3} \pi r^2 h$.
• Sphere Volume: $\frac{4}{3} \pi r^3$.