Enlargements and Constructions (Leaving Cert Mathematics): Revision Notes
Constructions
Introduction and equipment
Geometric construction involves drawing accurate shapes and lines using only a compass, straightedge (ruler without measurements), and protractor. These traditional tools allow mathematicians to create precise geometric figures based on mathematical properties rather than measurements.
Essential Equipment for Leaving Cert constructions:
- Compass for drawing arcs and circles
- Straightedge for drawing straight lines
- Protractor for measuring angles
- Set square for drawing parallel and perpendicular lines
Critical Exam Tip: When using a compass, you must leave the construction arcs visible as evidence that you used the correct method. This is essential for full marks in examinations.
Basic construction techniques
Constructing an angle of 60°
This construction uses the fundamental property that all angles in an equilateral triangle equal 60°. The method exploits the relationship between equal sides and equal angles in triangles.
Worked Example: Constructing a 60° Angle
Method:
- Draw a line segment XY
- Set your compass to the length of XY
- Place the compass point at X and draw an arc
- Keep the same radius, place the compass at Y and draw another arc
- The arcs intersect at point Z
- Join X to Z - angle YXZ = 60°
The triangle XYZ formed is equilateral because all three sides are equal (they're all the same radius length), so each angle must be 60°.
Constructing a tangent to a circle at a given point
A tangent to a circle is a straight line that touches the circle at exactly one point. The key property is that the tangent is perpendicular to the radius at the point of contact.
Worked Example: Constructing a Tangent to a Circle
Method:
- You're given a circle with centre O and a point X on the circumference
- Draw a line from the centre O to point X (this is a radius)
- Place a set square along the line OX
- Slide the set square until it reaches point X
- Draw a line through X perpendicular to OX
This line is the tangent to the circle at point X.
Constructing a parallelogram
A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. Understanding this fundamental property is essential for the construction process.
Worked Example: Constructing a Parallelogram
Method (when given two side lengths and one angle):
- Draw the base AB with the given length
- Use a protractor to construct the given angle at point A
- Mark the second given length along this line to find point D
- Use a set square to draw a line through D parallel to AB
- Use a compass to mark the length of AB along this parallel line to find point C
- Join BC to complete the parallelogram
Fundamental bisector constructions
The perpendicular bisector of a line segment
The perpendicular bisector of a line segment passes through the midpoint of the segment and is perpendicular to it. Every point on the perpendicular bisector is equidistant from the endpoints of the original segment.
Worked Example: Constructing a Perpendicular Bisector
Method:
- Draw line segment AB
- Set your compass to more than half the length of AB
- Place the compass at A and draw arcs above and below the line
- Keep the same radius, place the compass at B and draw arcs that intersect the first arcs
- Join the intersection points P and Q with a straight line
- This line PQ is the perpendicular bisector of AB
The bisector of an angle
The angle bisector divides an angle into two equal parts. Every point on the angle bisector is equidistant from the two arms of the angle.
Worked Example: Constructing an Angle Bisector
Method:
- Place your compass at the vertex of the angle
- Draw an arc that crosses both arms of the angle
- Place the compass at each intersection point in turn
- Draw two arcs that intersect inside the angle
- Draw a straight line from the vertex through this intersection point
Advanced triangle constructions
Understanding the relationship between different triangle centres is crucial for advanced geometric constructions. Each centre has unique properties that determine its construction method.
Circumcircle and incircle concepts
Circumcircle: The circle that passes through all three vertices of a triangle. Its centre is called the circumcentre.
Incircle: The largest circle that can fit inside a triangle, touching all three sides. Its centre is called the incentre.
Key Properties to Remember:
- The circumcentre is equidistant from all three vertices
- The incentre is equidistant from all three sides
- These properties determine the construction methods for each centre
How to construct the circumcircle of a triangle
The circumcentre is the point equidistant from all three vertices. It's found where the perpendicular bisectors of the triangle's sides meet.
Worked Example: Constructing the Circumcircle
Method:
- Draw triangle XYZ
- Construct the perpendicular bisector of side XY
- Construct the perpendicular bisector of side XZ
- These bisectors intersect at point O (the circumcentre)
- Set your compass to the distance from O to any vertex
- Draw a circle with centre O - this passes through all three vertices
How to construct the incircle of a triangle
The incentre is equidistant from all three sides of the triangle. It's found where the angle bisectors meet.
Worked Example: Constructing the Incircle
Method:
- Draw triangle XYZ
- Construct the bisector of angle XYZ
- Construct the bisector of angle XZY
- These bisectors intersect at point I (the incentre)
- Draw a perpendicular from I to any side of the triangle
- Set your compass to this perpendicular distance
- Draw a circle with centre I - this touches all three sides
How to construct the centroid of a triangle
The median of a triangle is a line from any vertex to the midpoint of the opposite side. The centroid is where all three medians intersect.
The centroid has a special property: it divides each median in the ratio 2:1, with the longer segment being closer to the vertex.
Worked Example: Constructing the Centroid
Method:
- Draw triangle XYZ
- Find the midpoint of side XZ by constructing its perpendicular bisector
- Draw the median from Y to this midpoint M
- Find the midpoint of side XY by constructing its perpendicular bisector
- Draw the median from Z to this midpoint N
- The medians YM and ZN intersect at point G (the centroid)
Worked example: Constructing a parallelogram
Worked Example: Complete Parallelogram Construction
Question: Construct parallelogram ABCD where AB = 6 cm, AD = 3.5 cm, and angle DAB = 50°.
Solution:
- Draw AB = 6 cm horizontally
- At point A, use a protractor to measure 50° from line AB
- Draw a line at this angle and mark point D so that AD = 3.5 cm
- Place a set square along AB and slide it to point D
- Draw a line through D parallel to AB
- Set compass to 6 cm and draw an arc from D to meet this parallel line at C
- Join BC to complete parallelogram ABCD
Applications in real life
Geometric constructions have practical applications beyond the classroom, demonstrating the real-world relevance of these mathematical techniques.
Real-World Applications:
- Architecture and engineering: Designing buildings with precise angles and measurements
- Map making: Finding equidistant points for optimal placement of facilities
- Art and design: Creating symmetrical and proportional designs
The perpendicular bisector property is particularly useful for finding a point equidistant from two locations - for example, determining the optimal location for a school that serves two towns equally.
Key Points to Remember:
- Always leave construction arcs visible - they're evidence of your method in exams
- Use a compass and straightedge only for pure constructions - measurements should come from compass settings, not ruler markings
- The circumcentre is equidistant from all vertices - found using perpendicular bisectors of sides
- The incentre is equidistant from all sides - found using angle bisectors
- Every point on a perpendicular bisector is equidistant from the endpoints of the original line segment
- Construction arcs are your evidence - never erase them in examination conditions