Geometry

The One Thing

Geometry uses precise definitions, angle and circle theorems, constructions, loci and proportional reasoning to describe shapes and calculate unknown measurements accurately. Diagrams should be analysed through established properties rather than appearance alone.

Definitions and Results

• Point, line and line segment: A point represents an exact position; a line extends infinitely in both directions; a line segment has two endpoints.
• Parallel lines: Lines that remain the same distance apart and never meet.
• Perpendicular lines: Lines that meet at a right angle of $90^{\circ }$.
• Acute, obtuse, reflex and right angles: An acute angle is less than $90^{\circ }$, an obtuse angle is between $90^{\circ }$ and $180^{\circ }$, a reflex angle is between $180^{\circ }$ and $360^{\circ }$, and a right angle is $90^{\circ }$.
• Angles on a straight line: Angles on a straight line add to $180^{\circ }$.
• Angles around a point: Angles around a point add to $360^{\circ }$.
• Vertically opposite angles: Opposite angles formed when two straight lines intersect are equal.
• Angles in parallel lines: Corresponding and alternate angles are equal; co-interior angles add to $180^{\circ }$.
• Triangle: A three-sided polygon whose interior angles add to $180^{\circ }$.
• Isosceles triangle: A triangle with two equal sides and two equal base angles.
• Equilateral triangle: A triangle with three equal sides and three angles of $60^{\circ }$.
• Polygon: A closed two-dimensional shape made from straight line segments.
• Regular polygon: A polygon with all sides equal and all interior angles equal.
• Interior angle sum of a polygon: For an $n$-sided polygon, the sum of the interior angles is $(n-2)\times 180^{\circ }$.
• Interior angle of a regular polygon: Each interior angle is $\frac{(n-2)\times 180^{\circ }}{n}$.
• Exterior angle of a regular polygon: Each exterior angle is $\frac{360^{\circ }}{n}$.
• Diagonals of a polygon: An $n$-sided polygon has $\frac{n(n-3)}{2}$ diagonals.
• Circle: A set of points that are all the same distance from a fixed centre.
• Radius and diameter: The radius is the distance from the centre to the circumference; the diameter passes through the centre and equals twice the radius.
• Chord: A line segment joining two points on a circle.
• Tangent: A line that touches a circle at exactly one point and is perpendicular to the radius at that point.
• Circumference and area of a circle:

$$C=2\pi r=\pi d,A=\pi r^2.$$

• Arc length: For central angle $\theta$,

$$\text{arc length}=\frac{\theta }{360^{\circ }}\times 2\pi r.$$

• Sector: A region enclosed by two radii and an arc. Its area is

$$\text{sector area}=\frac{\theta }{360^{\circ }}\times \pi r^2.$$

• Circle theorems: The angle in a semicircle is $90^{\circ }$; angles in the same segment are equal; opposite angles in a cyclic quadrilateral add to $180^{\circ }$; and the angle between a tangent and chord equals the angle in the alternate segment.
• Additional circle results: The radius to a tangent is perpendicular to the tangent. The perpendicular from the centre of a circle to a chord bisects the chord. The angle at the centre is twice the angle at the circumference standing on the same arc.
• Geometric construction: An accurate drawing made using mathematical instruments such as a ruler, compass and protractor.
• Perpendicular bisector: A line that meets a segment at $90^{\circ }$ and divides it into two equal parts. Every point on it is equidistant from the segment’s endpoints.
• Angle bisector: A line that divides an angle into two equal angles.
• Locus: The set of all points satisfying a given condition.
• Common loci: Points a fixed distance from a point form a circle; points equidistant from two points lie on the perpendicular bisector of the segment joining them; points equidistant from two intersecting lines lie on their angle bisectors.
• Similar shapes: Shapes with equal corresponding angles and proportional corresponding sides.
• Scale factor: The ratio of a corresponding length in an enlarged or reduced shape to the original length.
• Similarity and measurements: If the length scale factor is $k$, the area scale factor is $k^2$ and the volume scale factor is $k^3$.
• Congruent shapes: Shapes with the same size and shape; their corresponding sides and angles are equal.
• Triangle angle results: Angles in a triangle add to $180^{\circ }$. The exterior angle of a triangle equals the sum of the two opposite interior angles.
• Quadrilateral angle result: Angles in a quadrilateral add to $360^{\circ }$.
• Similarity tests: Two triangles are similar if they have two equal corresponding angles, by angle-angle similarity. Other tests require proportional corresponding sides with an included equal angle, or all corresponding sides proportional.
• Exact values and rounding: Use exact values involving $\pi$ when appropriate, and round only at the final stage unless instructed otherwise.

Worked Methods

Finding unknown angles

• Identify the relevant established angle property.
• Use angles on a straight line, angles around a point or vertically opposite angles where appropriate.
• If parallel lines are present, use equal corresponding or alternate angles, or supplementary co-interior angles.
• In a triangle, use the angle sum of $180^{\circ }$.
• In a quadrilateral, use the angle sum of $360^{\circ }$.
• For a triangle exterior angle, equate it to the sum of the two opposite interior angles.

Working with polygons

• Identify the number of sides, $n$.
• Calculate the interior angle sum:

$$(n-2)\times 180^{\circ }.$$

• For a regular polygon, divide by $n$ to find each interior angle:

$$\frac{(n-2)\times 180^{\circ }}{n}.$$

• Alternatively, calculate each exterior angle:

$$\frac{360^{\circ }}{n}.$$

• If diagonals are required, use:

$$\frac{n(n-3)}{2}.$$

Calculating circle measurements

• Identify the radius $r$, diameter $d$, and, where relevant, central angle $\theta$.
• Use the circumference formula:

$$C=2\pi r=\pi d.$$

• Use the area formula:

$$A=\pi r^2.$$

• For an arc, calculate the fraction $\frac{\theta }{360^{\circ }}$ of the full circumference:

$$\text{arc length}=\frac{\theta }{360^{\circ }}\times 2\pi r.$$

• For a sector, calculate the same fraction of the full circle area:

$$\text{sector area}=\frac{\theta }{360^{\circ }}\times \pi r^2.$$

• Retain exact values involving $\pi$ where appropriate and round only at the final stage unless instructed otherwise.

Applying circle theorems

• Identify whether the diagram contains a semicircle, the same segment, a cyclic quadrilateral, a tangent, a chord, a radius or a chord perpendicular from the centre.
• Apply the relevant theorem:

- angle in a semicircle $=90^{\circ }$; - angles in the same segment are equal; - opposite angles in a cyclic quadrilateral add to $180^{\circ }$; - the angle between a tangent and chord equals the angle in the alternate segment; - the radius to a tangent is perpendicular to the tangent; - the perpendicular from the centre to a chord bisects the chord; - the angle at the centre is twice the angle at the circumference standing on the same arc.

• Combine the result with triangle or quadrilateral angle sums if further calculation is required.

Constructing a perpendicular bisector

• Place the compass point at one endpoint of the segment.
• Draw arcs above and below the segment using a radius greater than half its length.
• Repeat from the other endpoint using the same compass width.
• Join the two arc intersections with a straight line.
• The resulting line meets the segment at $90^{\circ }$ and divides it into two equal parts.
• Every point on this line is equidistant from the segment’s endpoints.

Constructing an angle bisector

• Place the compass point at the angle’s vertex and draw an arc crossing both arms.
• From each crossing point, draw arcs of equal radius inside the angle.
• Join the vertex to the intersection of these arcs.
• The resulting line divides the angle into two equal angles.

Constructions should show clear arcs and accurate intersections rather than relying only on measurement.

Constructing loci

• Translate the verbal condition into a geometric condition.
• For points a fixed distance from a point, draw a circle centred at that point with the given radius.
• For points equidistant from two points, construct the perpendicular bisector of the segment joining them.
• For points equidistant from two intersecting lines, construct the relevant angle bisectors.
• Identify any required region or intersection of loci.

Using similarity and scale factors

• Match corresponding angles and sides correctly.
• Confirm that corresponding angles are equal and corresponding sides are proportional.
• For triangles, use angle-angle similarity if two corresponding angles are equal, or use the other stated similarity tests.
• Calculate the length scale factor:

$$k=\frac{\text{corresponding length in the new shape}}{\text{original corresponding length}}.$$

• Use proportional corresponding lengths to find unknown sides.
• Apply the appropriate measurement scale factor:

$$\text{length factor}=k,\text{area factor}=k^2,\text{volume factor}=k^3.$$

Dividing diagrams into familiar shapes

• Divide the diagram into familiar triangles, quadrilaterals, sectors or segments.
• Apply the relevant angle properties or formulas to each part.
• Add or subtract the resulting lengths, angles or areas.
• Check that the parts correspond to the required region of the original diagram.

Where It Goes Wrong

• Treating a diagram’s appearance as evidence instead of applying established angle, shape or circle properties.
• Forgetting that corresponding and alternate angles are equal in parallel lines, while co-interior angles add to $180^{\circ }$.
• Using the wrong polygon formula: the interior angle sum is $(n-2)\times 180^{\circ }$, whereas each exterior angle of a regular polygon is $\frac{360^{\circ }}{n}$.
• Confusing radius and diameter, despite the condition that the diameter passes through the centre and equals twice the radius.
• Omitting the degree fraction $\frac{\theta }{360^{\circ }}$ when calculating an arc length or sector area.
• Matching non-corresponding parts in similar shapes or using $k$ instead of $k^2$ for areas and $k^3$ for volumes.

What Gets Asked

• Define or distinguish geometric terms, including point, line segment, parallel, perpendicular, chord, tangent, locus, similarity and congruence.
• Calculate unknown angles using straight-line, around-a-point, vertically opposite, parallel-line, triangle, quadrilateral, polygon and circle theorems.
• Find polygon interior angles, exterior angles, interior angle sums and numbers of diagonals.
• Calculate the circumference, area, arc length or sector area of a circle.
• Apply circle theorems involving semicircles, cyclic quadrilaterals, tangents, chords, radii, central angles and angles in the same segment.
• Complete or interpret constructions of perpendicular bisectors and angle bisectors.
• Construct and describe loci for fixed distances, equal distances from two points, and equal distances from two intersecting lines.
• Determine whether shapes or triangles are similar or congruent and calculate missing corresponding lengths.
• Apply length, area and volume scale factors.
• Divide complex diagrams into familiar triangles, quadrilaterals, sectors or segments and combine the resulting measurements.