Constructions 2 (Edexcel GCSE Maths): Revision Notes
Constructions 2
All of these constructions are essential knowledge for your GCSE exam. You'll need to master using only a ruler and compasses for these geometric constructions.
Constructing triangles with given side lengths
When you need to construct a triangle with specific side measurements, following a systematic approach ensures accuracy and success every time.
A systematic construction method helps prevent errors and ensures your triangle will have the exact measurements required. Always work methodically through each step.
Method:
- Draw one side using your ruler and label it with the correct measurement
- Set your compasses to the length of the second side
- Draw an arc from one end of your base line
- Set your compasses to the length of the third side
- Draw an arc from the other end of your base line
- Mark where the arcs intersect - this gives you the third vertex
- Connect this point to both ends of your base line
Worked Example: Constructing a Triangle
To construct a triangle with sides 3cm, 4cm and 5.5cm:
Step 1: Draw the 5.5cm base line first (always start with the longest side) Step 2: Set compasses to 3cm and draw an arc from one end Step 3: Set compasses to 4cm and draw an arc from the other end Step 4: Mark where the arcs intersect to find the third vertex Step 5: Connect this point to both ends of your base line
Constructing angle bisectors
An angle bisector is a line that divides any angle into two equal parts. This construction technique works for any angle size and is fundamental to many geometric problems.
Method:
- Place your compass point at the vertex of the angle
- Draw arcs that cross both arms of the angle at equal distances from the vertex
- From each point where the arcs cross the arms, draw new arcs of equal radius
- These arcs will intersect at a point
- Draw a straight line from the vertex through this intersection point
- This line bisects your original angle
The equal distances and equal radii are crucial - this ensures the bisector is perfectly accurate. Any variation in these measurements will result in an inaccurate construction.
Constructing specific angles
Constructing a 45° angle
To create a 45° angle, you need to combine two construction techniques: perpendicular bisector construction and angle bisection.
Method:
- Draw a straight line and mark point P on it
- Construct the perpendicular bisector of this line at point P (this creates a 90° angle)
- Mark the midpoint M where the bisector meets your original line
- Set your compasses to distance PM
- Draw an arc from your bisector line
- Connect point P to where this arc intersects with a ruler
- This creates your 45° angle
This method works because you're essentially bisecting a 90° angle, and 90° ÷ 2 = 45°.
Constructing a 60° angle
A 60° angle comes from the properties of an equilateral triangle, where all sides are equal and all angles are 60°.
Method:
- Draw a base line and mark point P
- Set your compasses to any convenient distance
- Draw an arc above your line from point P
- Keep the same compass setting
- Draw another arc from where the first arc meets your base line
- Connect point P to where these arcs intersect
- This gives you a perfect 60° angle
Creating compound angles
You can create other angles by combining basic constructions. This technique allows you to construct any angle that's a multiple of 15°.
Worked Example: Constructing a 30° Angle
Step 1: First construct a 60° angle using the equilateral triangle method Step 2: Then construct the bisector of this 60° angle Step 3: The bisector creates two 30° angles
Why this works: Since 60° ÷ 2 = 30°, bisecting a 60° angle gives you the 30° you need.
Practice techniques
Mastering these construction techniques requires understanding the key principles behind each method.
For angle bisector questions:
- Always start by marking equal distances from the vertex on both arms
- Use the same compass radius for the intersecting arcs
- Keep your compass settings consistent throughout
For triangle construction:
- Always draw your longest side first as the base - this makes the construction more stable
- Double-check your compass settings match the required measurements
- Make light construction marks that you can erase later
Key Points to Remember:
- All constructions use only ruler and compasses - no protractors or other measuring tools
- Equal distances and equal radii are the key principles for accurate bisectors
- Start with the longest side when constructing triangles
- 60° angles come from equilateral triangles - all angles in an equilateral triangle are 60°
- Combine basic constructions to create other angles (like bisecting 60° to get 30°)