Euclidean Geometry-G11-Maths-T4P2

Euclidean Geometry

Euclidean geometry questions require accurate diagrams, logical reasoning and the correct geometrical reasons. In examinations, every statement used to calculate or prove an angle must be supported by a valid reason. Do not assume that lines are equal, parallel or perpendicular simply because they appear so in the diagram.

Circle geometry

A circle is the set of all points in a plane that are the same distance from a fixed point called the centre.

Important terms include:

Radius: a line segment from the centre to the circumference.
Diameter: a chord passing through the centre.
Chord: a line segment joining two points on the circumference.
Arc: part of the circumference between two points.
Tangent: a line touching a circle at one point.
Secant: a line passing through a circle at two points.
Cyclic quadrilateral: a quadrilateral with all four vertices on the circumference of a circle.

All radii of the same circle are equal. Therefore, triangles formed by two radii are isosceles triangles.

Angles in circles

The angle at the centre of a circle is twice the angle at the circumference when both angles are subtended by the same chord or arc.

∠ A O B = 2 ∠ A C B

The correct reason is: angle at centre equals twice angle at circumference.

Example

Chord AB subtends an angle of 108 degrees at centre O. Calculate the angle subtended by the same chord at point C on the major arc.

∠ A C B = \frac{1}{2} ∠ A O B

∠ A C B = \frac{1}{2} (108^{°})

∠ A C B = 54^{°}

An angle subtended by a diameter at the circumference is 90 degrees. This is often described as the angle in a semicircle.

∠ A C B = 90^{°}

The correct reason is: angle in a semicircle.

Angles subtended by the same chord in the same segment are equal.

∠ A C B = ∠ A D B

The correct reason is: angles in the same segment.

Chords

Equal chords subtend equal angles at the centre of the same circle.

If chords AB and CD are equal, then:

A B = C D

∠ A O B = ∠ C O D

The converse is also true: chords that subtend equal angles at the centre are equal.

Equal chords are also the same perpendicular distance from the centre. Conversely, chords that are the same distance from the centre are equal.

Remember that the diameter is the longest chord in a circle.

Example

Two equal chords subtend angles at the centre. One angle is given as 72 degrees. Calculate the other angle.

Since equal chords subtend equal angles at the centre:

∠ A O B = ∠ C O D

∠ C O D = 72^{°}

Tangents

A tangent touches a circle at exactly one point, called the point of contact.

Tangents drawn from the same external point are equal in length. If PA and PB are tangents from P to a circle, then:

P A = P B

The correct reason is: tangents from a common point are equal.

This creates an isosceles triangle, so the angles opposite the equal tangents are equal.

Example

PA and PB are tangents to a circle. If PA is 12 cm, determine PB.

P A = P B

P B = 12 cm

The reason is: tangents from a common point.

The tangent-chord theorem states that the angle between a tangent and a chord equals the angle in the alternate segment subtended by that chord.

If PT is a tangent at A and AB is a chord, then:

∠ P A B = ∠ A C B

The correct reason is: tangent-chord theorem.

Cyclic quadrilaterals

A quadrilateral is cyclic if all four vertices lie on the circumference of the same circle.

The opposite angles of a cyclic quadrilateral are supplementary.

∠ A + ∠ C = 180^{°}

∠ B + ∠ D = 180^{°}

The correct reason is: opposite angles of a cyclic quadrilateral are supplementary.

Example

ABCD is a cyclic quadrilateral and angle A is 74 degrees. Calculate angle C.

∠ A + ∠ C = 180^{°}

74^{°} + ∠ C = 180^{°}

∠ C = 106^{°}

An exterior angle of a cyclic quadrilateral equals the interior opposite angle.

∠ D C E = ∠ D A B

The correct reason is: exterior angle of a cyclic quadrilateral equals the interior opposite angle.

The converse is useful when proving that a quadrilateral is cyclic. A quadrilateral is cyclic if:

A pair of opposite angles is supplementary.
An exterior angle equals the interior opposite angle.
Two angles subtended by the same line segment are equal.

Angles subtended by chords and arcs

The endpoints of a chord define two arcs. An angle subtended by a chord depends on the segment in which the angle lies.

Angles subtended by the same chord on the same side of the chord are equal.

∠ A C B = ∠ A D B

Angles subtended by the same chord on opposite sides of the chord are supplementary.

∠ A C B + ∠ A D B = 180^{°}

A larger arc generally subtends a larger angle at the centre. A semicircular arc subtends a straight angle at the centre and a right angle at the circumference.

∠ A O B = 180^{°}

∠ A C B = 90^{°}

Perpendicular from the centre to a chord

The perpendicular drawn from the centre of a circle to a chord bisects the chord.

If OM is perpendicular to chord AB, then:

O M ⊥ A B

A M = M B

The converse is also true: a line drawn from the centre to the midpoint of a chord is perpendicular to the chord.

Example

The radius of a circle is 10 cm. The perpendicular distance from the centre to a chord is 6 cm. Calculate the length of the chord.

The perpendicular from the centre bisects the chord. Let M be the midpoint of chord AB. Apply the theorem of Pythagoras in right-angled triangle OMA.

{O A}^{2} = {O M}^{2} + {A M}^{2}

10^{2} = 6^{2} + {A M}^{2}

100 = 36 + {A M}^{2}

{A M}^{2} = 64

A M = 8 cm

Since the chord is bisected:

A B = 2 (A M)

A B = 2 (8)

A B = 16 cm

Radius and tangent relationships

A radius drawn to the point of contact is perpendicular to the tangent.

If PT is tangent to a circle at T and O is the centre, then:

O T ⊥ P T

∠ O T P = 90^{°}

The correct reason is: radius perpendicular to tangent.

The converse is also true. If a line is perpendicular to a radius at the point where the radius meets the circle, the line is a tangent to the circle.

Example

OT is a radius of length 5 cm and PT is a tangent of length 12 cm. Calculate OP.

Since a radius is perpendicular to a tangent at the point of contact, triangle OTP is right-angled.

{O P}^{2} = {O T}^{2} + {P T}^{2}

{O P}^{2} = 5^{2} + 12^{2}

{O P}^{2} = 25 + 144

{O P}^{2} = 169

O P = 13 cm

Proofs of theorems

Proof: A perpendicular from the centre bisects a chord

Let O be the centre, AB be a chord and OM be perpendicular to AB. Join OA and OB.

Since OA and OB are radii:

O A = O B

OM is a common side:

O M = O M

Both angles at M are right angles:

∠ O M A = ∠ O M B = 90^{°}

Therefore, the right-angled triangles are congruent by RHS.

△ O M A \equiv △ O M B

Corresponding sides of congruent triangles are equal.

A M = M B

Therefore, OM bisects chord AB.

Proof: Opposite angles of a cyclic quadrilateral are supplementary

Let ABCD be a cyclic quadrilateral with centre O. The angle at the circumference equals half the angle at the centre subtended by the same arc.

Angles A and C together subtend the complete circumference. The angles at the centre therefore add to 360 degrees.

∠ A + ∠ C = \frac{1}{2} (360^{°})

∠ A + ∠ C = 180^{°}

Therefore, opposite angles of a cyclic quadrilateral are supplementary.

Applying theorems to solve geometry problems

Read the information marked on the diagram before starting. Identify radii, equal tangents, chords, diameters and cyclic quadrilaterals.

Example

AB is a diameter of a circle. C is a point on the circle and a tangent is drawn at A. If angle ABC is 38 degrees, calculate angle BAC and the angle between the tangent and chord AC.

Since AB is a diameter:

∠ A C B = 90^{°}

Using the sum of angles in a triangle:

∠ B A C + ∠ A B C + ∠ A C B = 180^{°}

∠ B A C + 38^{°} + 90^{°} = 180^{°}

∠ B A C = 52^{°}

By the tangent-chord theorem, the angle between the tangent and chord AC equals the angle subtended by chord AC in the alternate segment.

∠ P A C = ∠ A B C

∠ P A C = 38^{°}

Exam Tip

Write one statement per line and give its reason immediately. Use only information given in the question or facts already proved. Never use the result that the question asks you to prove as a reason.

Similarity and proportional relationships where applicable

Triangles are similar when they are equiangular. Circle theorems are often used to prove that two pairs of corresponding angles are equal.

If two triangles are similar, their corresponding sides are proportional.

△ A B C \sim △ D E F

\frac{A B}{D E} = \frac{B C}{E F} = \frac{A C}{D F}

Example

Triangles ABC and DBC have two pairs of equal angles and are therefore similar. Suppose the corresponding sides are AB and DB, AC and DC, and BC and BC.

△ A B C \sim △ D B C

Therefore:

\frac{A B}{D B} = \frac{A C}{D C} = \frac{B C}{B C}

If AB is 6 cm, DB is 9 cm and AC is 8 cm, calculate DC.

\frac{A B}{D B} = \frac{A C}{D C}

\frac{6}{9} = \frac{8}{D C}

6 (D C) = 9 (8)

6 D C = 72

D C = 12 cm

Common Mistake

Do not state that triangles are similar merely because they look alike. Prove similarity by showing that corresponding angles are equal. Maintain the correct order of corresponding vertices when writing the similarity statement and all proportions.

Study Guide For Grade

11

Mathematics