Leaving Cert Geometry

Constructions: Instruments allowed: Straight-edge; pencil; compass; ruler; protractor; set-squares with 90o, 30o, 45o, 60o angles.

(supported by 46 definitions; 20 propositions; 5 axioms; 21 theorems; 6 Corollaries)

You can use the special Student Disc  Part I (J. Cert/L.Cert) and Part II (L.Cert.)  produced by Project Maths.

F

(B)

O

(G)

H

(A)

1

Bisector of any given angle, using only compass and straight edge.

2

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

3

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

4

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

5

Line parallel to a given line, through given point.

6

Division of a segment into 2 or 3 equal segments without measuring it.

7

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

8

Line segment of a given length on a given ray.

9

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

10

Triangle, given lengths of three sides (SSS data).

11

Triangle, given SAS data.

12

Triangle, given ASA data.

13

Right angled triangle, given the length of the hypotenuse and one other side.(RHS)   (Right-angle).

14

Right-angled triangle, given one side and one of the acute angles (several cases).[Given Hypotenuse]

15

Rectangle given side lengths.

16

Construct the circumcentre and circumcircle of a given triangle, using only straight edge and compass

17

Construct the incentre and incircle of a given triangle, using only straight edge and compass.

18

Construct an angle of 60˚ without using a protractor or set square.

19

Construct the tangent to a given circle at a given point on it.

20

Construct a parallelogram, given the length of the sides and the measure of the angles

21

Construct the centroid of a triangle

22

Construct the orthocentre of a triangle.

Outside the  course: Other ideas to look at
  1. Impossible Constructions Trisecting an angle with straight-edge and compass only was proved to be impossible in 1837 by Pierre Laurent Wantzel (1814 – 1848).
  2.    Everiste Galois (1811-1832) outlined the theory of groups in 1831 shortly before he died. Using Galois theory it becomes easy to answer the classical construction problems such as the problem of trisecting an angle with straight-edge and compass only although Galois himself did not provide such an answer.

 

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