Geometry

This very broad topic is one of the most ancient areas of study known to man. It may not be hugely inviting for many students, but it is not possible to avoid. Included in this vast area are points, angles, lines, triangles, parallelograms, etc. - the list is indeed long.

Euclid is the point where our study of modern day geometry really begins. No questions will be set on him directly but we will use many of his approaches. There are two full questions on the ordinary level paper and three on the higher-level paper.

Key topics

Below is a list of all the key topics within this section of the Junior Cert syllabus. Areas of interest to higher-level students only have been highlighted.

For an interactive explanation of difficult concepts and the opportunity to solve practical problems within this section, consult our Geometry lessons.

• Points, Lines, Angles
• Types of angles and properties of angles
• Distance from a point to a line
• Parallel lines
• Perpendicular lines
• Triangles and types of triangles
• The sum of the angles in a triangle equals 180°
• The exterior angle of a triangle equals the sum of the interior opposite angles
• Quadrilateral and types of quadrilateral
• Circle: centre, radius, diameter
• Chord, secant, segment, tangent
• The angle in a semicircle is a right angle
• A tangent to a circle is perpendicular to a radius at the point of contact
• The opposite angles in a cyclic quadrilateral add up to 180°
• Pythagoras' Theorem
• Constructions, perpendicular bisector, bisector of an angle
• Constructing triangles
• Congruent triangles
• Transformations
• Translation
• Axis of symmetry
• Axial symmetry
• Central symmetry
• Coordinate geometry
• Distance formula
• Midpoint formula
• Slope of a line
• Equation of a line
• Graphing lines
• Point of intersection of two lines Higher level only
• Slope of a line from its equation
• Lines parallel and perpendicular to a given line
• Lines containing the origin
• Lines parallel to the axes
• Transformations on the coordinated plane
• Using congruent triangles to prove propositions
• Similar triangles
• More difficult constructions
  Proofs of the 10 theorems [Higher level only]
  
  1. Vertically opposite angles are equal in measure.
  2. The measure of the three angles of a triangle sum to 180°.
  3. The exterior angle in a triangle equals the sum of the interior opposite angles.
  4. If two sides of a triangle are equal in measure, then the sides opposite these angles are also equal in measure.
  5. Opposite sides and opposite angles of a parallelogram are equal in measure.
  6. A diagonal bisects the area of a parallelogram.
  7. The measure of the angle at the centre of a circle is twice the measure of the angle at the circumference, standing on the same arc.

• Deduction one: all angles on the same arc are equal.
• Deduction two: the angle in a semicircle is a right angle. Link to lesson
• Deduction three: the sum of the opposite angles in a cyclic quadrilateral is 180°.

8. A line through the centre of a circle perpendicular to a chord bisects the chord.
9. If two triangles are equiangular, the lengths of corresponding sides are in proportion.
10. In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Sample questions (from official D.E.S. sample exam papers)

Ordinary level

Solution Triangle abd and triangle adc.

Higher level