# Proofs, Derivations and Constructions for Leaving Cert Higher Level Maths

This page gives a comprehensive list of all proofs, derivations and constructions that are on the leaving cert higher level maths syllabus. I have taken this information directly from the syllabus to be as accurate as possible. The complete syllabus can be found here.

Where a video solution of the proof is available simply click on the   Video  button to go to the video

Proofs

• prove theorems 11,12,13,

• prove that √2 is not rational  Video

• prove De Moivre’s Theorem by induction for n ∈ N  Video

• prove by induction

• simple identities such as the sum of the first n natural numbers and the sum of a finite geometric series

• simple inequalities such as n! > 2n , 2n ≥ n2 (n ≥ 4) (1+ x) n ≥ 1+nx (x > –1)

• factorisation results such as 3 is a factor of 4n –1

Derivations

• derive the trigonometric formulae 1 Video, 2, 3, 4 Video, 5 Video, 6, 7, 9

• derive the Sine Rule Formula (trig formula 2) Video

• derive the Cosine Rule Formula (trig formula 3)Video

• derive the formula for the sum to infinity of geometric series by considering the limit of a sequence of partial sums

• solve problems involving finite and infinite geometric series including applications such as recurring decimals and financial applications, e.g. deriving the formula for a mortgage repayment

Constructions

• geometrically construct √2 and √3

• perform constructions 16-22 (1-15 known from JC - A knowledge of the constructions prescribed for JC-HL will be assumed, and may be examined)

• The prescribed constructions are:

• 1. Bisector of a given angle, using only compass and straight edge.   Video

• 2. Perpendicular bisector of a segment, using only compass and straight edge.   Video

• 3. Line perpendicular to a given line l, passing through a given point not on l.   Video

• 4. Line perpendicular to a given line l, passing through a given point on l.   Video

• 5. Line parallel to given line, through given point.

• 6. Division of a segment into 2, 3 equal segments, without measuring it.

• 7. Division of a segment into any number of equal segments, without measuring it.

• 8. Line segment of given length on a given ray.

• 9. Angle of given number of degrees with a given ray as one arm.

• 10. Triangle, given lengths of three sides.

• 11. Triangle, given SAS data.

• 12. Triangle, given ASA data.

• 13. Right-angled triangle, given the length of the hypotenuse and one other side.

• 14. Right-angled triangle, given one side and one of the acute angles (several cases).

• 15. Rectangle, given side lengths.

• 16. Circumcentre and circumcircle of a given triangle, using only straightedge and compass.  Video

• 17. Incentre and incircle of a given triangle, using only straight-edge and compass.  Video

• 18. Angle of 60◦ , without using a protractor or set square.  Video

• 19. Tangent to a given circle at a given point on it.  Video

• 20. Parallelogram, given the length of the sides and the measure of the angles.

• 21. Centroid of a triangle.  Video

• 22. Orthocentre of a triangle.

Accessed on 20/04/2020