Constructions 1 (Edexcel GCSE Maths): Revision Notes
Constructions 1
Constructing perpendicular lines is a fundamental skill in geometry that you'll need to master for your GCSE exam. There are three main methods you might be asked to use, all requiring only a ruler and compass.
What are perpendicular lines?
Perpendicular lines meet at exactly 90 degrees (right angles). When you construct these lines, you're creating precise geometric relationships using only basic tools.
Understanding perpendicular lines is essential for many geometric constructions and will appear frequently in GCSE questions.
Method 1: Perpendicular through a point on the line
This method creates a perpendicular line that passes through a specific point on an existing line.
Worked Example: Perpendicular Through Point P
Step 1: Place your compass point on the given point P on the line
Step 2: Draw arcs on both sides of point P, marking two points on the line at equal distances
Step 3: Keep your compass setting the same
Step 4: From each of these new points, draw arcs above and below the line
Step 5: Where these arcs intersect, draw a straight line through point P
Result: This line is perpendicular to the original line
The key principle here is using equal distances from your starting point to ensure accuracy.
Method 2: Perpendicular from a point not on the line
This method constructs a perpendicular from a point that sits away from the line, dropping down to meet it at right angles.
Worked Example: Perpendicular From External Point P
Step 1: From point P (not on the line), use your compass to mark two points on the line at equal distances
Step 2: The compass should be set wide enough so the arc crosses the line in two places
Step 3: Keep the same compass setting
Step 4: From each intersection point on the line, draw arcs on the opposite side from point P
Step 5: Where these arcs meet, draw a line from point P through this intersection
Result: This creates your perpendicular line
This method ensures the perpendicular meets the line at the shortest possible distance from your point.
Method 3: Perpendicular bisector of a line segment
A perpendicular bisector cuts a line segment exactly in half at right angles.
Worked Example: Perpendicular Bisector of Line AB
Step 1: Set your compass to more than half the length of line segment AB
Step 2: Place the compass point at A and draw arcs above and below the line
Step 3: Without changing the compass setting, place it at point B
Step 4: Draw arcs above and below the line to intersect with your first arcs
Step 5: Draw a straight line through both intersection points
Result: This line bisects AB perpendicularly
The perpendicular bisector has a special property - every point on it is equidistant from both ends of the original line segment.
Exam guidance and tips
Understanding what examiners look for is crucial for achieving full marks in construction questions.
Examiners' advice: Students often lose marks by not showing their construction lines. To get full marks, you must be accurate and show all your construction lines.
Common exam requirements:
- Questions are typically worth 2 marks
- You must use only ruler and compass (no protractor)
- All construction arcs must be visible
- The final perpendicular line should be clearly drawn
Essential exam technique:
- Never rub out your construction lines
- Keep your compass setting consistent throughout each method
- Draw arcs long enough to create clear intersection points
- Use a sharp pencil for precision
Constructions checklist
Before submitting your work, ensure you have:
- Used good compasses with stiff arms
- Used a sharp pencil
- Used a transparent ruler
- Marked any angles clearly
- Labelled any lengths as required
- Shown all construction lines visibly
Key Points to Remember:
- All three methods use the principle of equal distances - your compass creates arcs at the same distance from key points
- Construction lines must always be shown - they're part of your working and earn you marks
- Accuracy is crucial - small errors in compass work lead to inaccurate perpendiculars
- Only ruler and compass are allowed - no protractors or angle measurers
- Perpendicular bisectors are special - they cut lines exactly in half at right angles