Paul Keogan

LCOL Geometry

Constructions, theorems and their proofs, angles, circle theorems, congruent and similar triangles, and transformations.

Your progress through this topic0 of 16 lessons

Lessons in this topic

All 16 lessons, in order — each one ticks off as you open it.

1Theorems, Axioms and Corollaries 2Constructions 3Constructing the circumcircle 4Constructing the incircle 5Constructing the Centroid 6Angles 7Circle Questions 8Congruent Triangles 9Similar Triangles 10Transformations 1 11Transformations 2 Enlargements 12Proof of Theorem 4 (Practise)13Proof of Theorem 6 (Practise)14Proof of Theorem 9 (Practise)15Proof of Theorem 14 (Practise)16Proof of Theorem 19 (Practise)

Key ideas in this topic

★ Must learn

Theorems on the course

There are 21 theorems on the Leaving Cert Ordinary Level course. At OL you recognise and apply them — you are not asked to prove the hard ones.

LCOL Geometry 1

★ Must learn

What is an axiom?

An axiom is a statement accepted as true without proof. There are 5 axioms; the one to learn is that a straight angle is 180°.

LCOL Geometry 1

★ Must learn

Corollaries on the course

A corollary is a small result that follows directly from a theorem. There are 6 corollaries (e.g. an angle in a semicircle is 90°).

LCOL Geometry 1

Translation

A slide — every point moves the same distance in the same direction.

LCOL Geometry 11

Axial symmetry

A flip in a line — the image is the mirror image in the axis.

LCOL Geometry 11

Central symmetry

A flip through a point — every point passes through the centre to the far side.

LCOL Geometry 11

Rotation

A turn about a point through an angle.

LCOL Geometry 11

Enlargement

A scaling by a factor from a centre — every length is multiplied by the scale factor.

LCOL Geometry 11

★ Must learn

Constructions on the course

There are 21 constructions on the Ordinary Level course, done with compass and straight edge only — measuring the answer with a protractor earns no marks; the arcs are the marks.

LCOL Geometry 2

Angles on the same arc

Two angles standing on the same arc (subtended by the same chord) are equal.

LCOL Geometry 7

★ Must learn

Proving triangles congruent

Two triangles are congruent (identical) if they match on SSS, SAS, ASA or RHS. Set the proof out in two columns, one triangle each.

LCOL Geometry 8

★ Must learn

Proving triangles similar

Two triangles are similar if their angles are equal and their sides are in proportion. Set up matching ratios to find a missing length.

LCOL Geometry 9

Paul Keogan

LCOL Area, Volume and Measurement

LCOL Area, Volume & Measurement

Area, volume and surface area — rectangles, circles and sectors, triangles, cylinders, spheres, cones, prisms and the trapezoidal rule.

Your progress through this topic0 of 9 lessons

Lessons in this topic

All 9 lessons, in order — each one ticks off as you open it.

1Rectangles, Squares, Rectangular blocks and Cubes 2Circles and Sectors 3Triangles, Parallelograms and trapeziums 4Unit Conversion 5Cylinders 6Spheres 7Cones 8Triangular Prism 9Trapezoidal Rule

Key ideas in this topic

Triangles, Parallelograms and trapeziumsLCOL Area, Volume & Measurement 3

CylindersLCOL Area, Volume & Measurement 5

SpheresLCOL Area, Volume & Measurement 6

Triangular PrismLCOL Area, Volume & Measurement 8

Trapezoidal RuleLCOL Area, Volume & Measurement 9

Paul Keogan

LCOL Co-Ordinate Geometry of the circle

LCOL The Circle

Co-ordinate geometry of the circle — its equation, points on/inside/outside, finding it from a diameter, tangents, and where a line meets a circle.

Your progress through this topic0 of 5 lessons

Lessons in this topic

All 5 lessons, in order — each one ticks off as you open it.

1Introduction 2Points on, inside or outside a circle 3Given the Diameter of a circle 4Equation of a tangent to a circle 5Point of Intersection of a line and a circle

Key ideas in this topic

Equation of a circle

centre (h, k), radius r

LCOL The Circle 1

Tangent ⟂ radius

At the point of contact, the tangent is perpendicular to the radius, so their slopes multiply to −1. Find the slope of the radius, then invert it and change the sign to get the tangent’s slope.

LCOL The Circle 4

Paul Keogan

LCOL Co-Ordinate Geometry of the Line

LCOL The Line

Co-ordinate geometry of the line — slope, midpoint, distance, the equation of a line, parallel and perpendicular lines, intersections and the area of a triangle.

Your progress through this topic0 of 13 lessons

Lessons in this topic

All 13 lessons, in order — each one ticks off as you open it.

1Plotting Points 2Rise over Run 3Slope Formula 4Midpoint Formula 5Distance between two points 6Finding the Equation of a Line 7Drawing a Line 8y= mx + c 9Parallel and Perpendicular lines 10Proving a point is on a line 11Point of Intersection of two lines 12Horizontal and Vertical Lines 13Area of a Triangle

Key ideas in this topic

★ Must learn

On the axes

On the x-axis, y = 0; on the y-axis, x = 0. Use this to find where a line crosses each axis.

LCOL The Line 1

Parallel lines

Parallel lines have equal slopes.

LCOL The Line 9

Perpendicular lines

Flip the fraction and change the sign.

LCOL The Line 9

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LCHL Maths – Co-ordinate Geometry of The Circle

LCHL Co-ordinate Geometry of the Circle

The LCHL Co-ordinate Geometry of the Circle strand — the equation of a circle, the general form, and tangents.

Your progress through this topic0 of 9 lessons

Lessons in this topic

All 9 lessons, in order — each one ticks off as you open it.

1Introduction 2Characteristics of the equation of a circle 3Proving a point is: on a circle, inside a circle or outside a circle 4Point of intersection of a line and a circle ( tangents, chords etc)5Intersecting circles/Touching circles 6Circles with the axes as a tangent 7The circle 7 Find the equation of the tangent to the circle at the point.8Find the equation of the tangent to the circle from the point.9Simultaneous Equations

Key ideas in this topic

Centre & radius from general formThe Circle

Methods you must know

Proving a line is a tangentone solution ⇒ tangent

Circles touching internallyd = |r₁ − r₂|

Check it against the exam

Jump straight to the fully worked solution for that paper.

2023 Paper 2 Q4 →2022 Paper 2 Q3 →2021 Paper 2 Q3 →2020 Paper 2 Q2 →2020 Paper 2 Q3 →

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LCHL Paper 2 Proofs

The Paper 2 proofs — the eight trigonometry proofs (Trigonometry 4) and the geometry theorem proofs. Theorems 11, 12 and 13 are the examinable ones; the older JC theorems (4, 6, 9, 14, 19) are awareness-only.

Your progress through this topic0 of 11 lessons

Proofs in this section

All 11 proofs, in order — each one ticks off as you open it.

1Trig Proof 1★ Learn off 2Trig proof 2★ Learn off 3Trig Proof 3★ Learn off 4Trig proof 4★ Learn off 5Trig proof 5★ Learn off 6Trig Proof 6★ Learn off 7Trig proof 7★ Learn off 8Trig proof 8★ Learn off 9Proof of theorem 11★ Learn off 10Proof of theorem 12★ Learn off 11Proof of theorem 13★ Learn off

Check it against the exam

Jump straight to the fully worked solution for that paper.

2023 Paper 1 Q8 →2023 Paper 2 Q2 →2023 Paper 2 Q4 →2022 Paper 2 Q6 →2021 Paper 2 Q4 →

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LCHL Maths – Area, Volume and Measurement

LCHL Area, Volume and Measurement

The LCHL Area, Volume and Measurement strand — areas and volumes of standard shapes, and approximating area with the trapezoidal rule.

Your progress through this topic0 of 8 lessons

Lessons in this topic

All 8 lessons, in order — each one ticks off as you open it.

1JC Revision 2Unit Conversion 3Trapezoidal rule 4Area of a triangle using Trigonometry 5Area of a triangle using co-ordinate geometry 6Area between a function and the x-axis (integration)7Area between a function and the y-axis (integration)8Differentiation rate problems

Key ideas in this topic

Cylinder — curved surface areathe wraparound only

Cylinder — total surface areacurved + the two circular ends

Volume of a coneh = the perpendicular height

Cone — curved surface areal = the SLANT height

Cone — total surface areacurved + the circular base

Cone — slant heightPythagoras links r, h and l

Surface area of a hemispherecurved + the flat circular face

Volume of a prismany consistent cross-section

Check it against the exam

Jump straight to the fully worked solution for that paper.

2015 Paper 1 Q3 →

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LCHL Maths – Geometry

LCHL Geometry

The LCHL Geometry strand — the synthetic-geometry theorems, constructions, and the examinable theorem proofs (Theorems 11, 12 and 13).

Your progress through this topic0 of 19 lessons

Lessons in this topic

All 19 lessons, in order — each one ticks off as you open it.

1Axioms, Theorems, Corollaries★ Learn off 2Construction 1 – 15 , 18, 19, 20★ Learn off 3Construction 16 – Circumcircle★ Learn off 4Construction 17 – Incircle★ Learn off 5Construction 21 – Centroid★ Learn off 6Construction 22 – Orthocentre★ Learn off 7Proof of Theorem 4★ Learn off 8Proof of theorem 6★ Learn off 9Proof of Theorem 9★ Learn off 10Proof of Theorem 14★ Learn off 11Proof of theorem 19★ Learn off 12Proof of theorem 11★ Learn off 13Proof of theorem 12★ Learn off 14Proof of theorem 13★ Learn off 15Questions on theorem 19 (circle questions)★ Learn off 16Similar Triangles★ Learn off 17Congruent Triangles★ Learn off 18Rotations, Translations, Central symmetry, Axial symmetry★ Learn off 19Enlargements★ Learn off

Key topics to learn off

There’s a lot here — learn the axioms, the theorem proofs, the constructions and the transformations.

5 axioms21 theorems6 corollaries22 constructions3 examinable proofs (Theorems 11, 12, 13)Centroid, circumcentre, orthocentreCongruent & similar trianglesTransformations & enlargements