In this chapter, you will learn
- —Master compass and straightedge construction techniques
- —Construct perpendicular bisectors and perpendiculars to a line
- —Construct angle bisectors and copy angles
- —Construct triangles using SSS, SAS, ASA, and RHS criteria
- —Apply scale factor concepts for scaled drawings
Introduction to Practical Geometry
Practical Geometry involves constructing geometric shapes using compass and straightedge (unmarked ruler). Unlike drawing, construction creates exact geometric figures.
Essential Tools:
- Compass: Used to mark equal distances and draw arcs/circles
- Straightedge: Unmarked ruler used to draw straight lines
- Protractor: For measuring and constructing angles (optional for basic constructions)
Key Principles:
- All constructions must be accurate to within 1-2 mm
- Compass should not be adjusted between related construction steps
- Arcs must be dark enough to mark intersection points clearly
- Show all construction lines and arcs (don't erase them)
Exam Tip
Always show your construction lines and arcs. Examiners check your method, not just the final figure.
Common Mistake
Adjusting the compass between steps causing incorrect measurements. Keep compass fixed for related constructions.
Perpendicular Bisector and Perpendiculars
A perpendicular bisector divides a line segment into two equal parts at 90°. A perpendicular is a line drawn at 90° to another line.
Construction of Perpendicular Bisector of a Line Segment:
- Given segment AB, open compass to more than half of AB
- From A, draw arcs above and below AB
- With same compass width, from B, draw arcs intersecting previous arcs at points P and Q
- Join P and Q - this is the perpendicular bisector
Construction of Perpendicular from Point to Line:
- Given line AB and external point P
- From P, draw arcs intersecting AB at points C and D
- From C and D, draw equal arcs intersecting at point Q
- Join PQ - this is perpendicular to AB
Uses: Creating right angles, finding midpoints, constructing squares and rectangles.
Exam Tip
For perpendicular bisector questions, remember the compass must be > half the segment. Show both arcs clearly.
Common Mistake
Opening compass less than half the segment results in arcs that don't intersect. Always estimate > half.
Angle Bisector and Angle Copying
An angle bisector divides an angle into two equal parts. Angle copying creates an identical angle at a different location.
Construction of Angle Bisector:
- Given angle ABC, place compass at B and draw an arc cutting BA and BC at points P and Q
- From P and Q, draw equal arcs intersecting at point R
- Join BR - this bisects angle ABC into two equal parts
Construction of Angle Equal to Given Angle:
- Given angle ABC, draw an arc from B cutting BA and BC at points P and Q
- Draw a ray from point X
- From X, draw an arc of same radius intersecting ray at point P'
- With compass set to PQ distance, from P' draw arc intersecting arc at Q'
- Join XQ' to form angle equal to ABC
Note: 60° and 120° angles can be constructed using compass only (without protractor).
Exam Tip
Angle bisector construction must show arcs from both arms of the angle. Make arcs dark and clear.
Common Mistake
Not making arcs from P and Q equal in size. This results in inaccurate bisection of the angle.
Construction of Triangles
Triangles can be constructed using different criteria based on given information:
SSS (Side-Side-Side): All three sides are given
Method: Draw base, use compass to mark point intersections for remaining vertices.
SAS (Side-Angle-Side): Two sides and included angle are given
Method: Draw base and angle, then mark second side length.
ASA (Angle-Side-Angle): Two angles and included side are given
Method: Draw base, construct both angles at endpoints, extend rays to meet.
RHS (Right-Hypotenuse-Side): Right angle, hypotenuse, and one side are given
Method: Draw hypotenuse, use perpendicular bisector, mark remaining side.
Important: AAA and SSA criteria do NOT uniquely determine a triangle.
Exam Tip
Always verify the given information matches one of SSS, SAS, ASA, or RHS. Show all construction lines.
Common Mistake
Using wrong construction method for given criteria. Carefully read what measurements are provided.
Scale Drawings and Similar Figures
A scale drawing is a proportional representation of an object at a smaller or larger size. Scale factor is the ratio of drawing dimension to actual dimension.
Scale Factor = Dimension in Drawing / Actual Dimension
Examples of Scales:
- Scale 1:100 means 1 cm on drawing = 100 cm actual (1 cm = 1 m)
- Scale 1:50,000 on map means 1 cm = 0.5 km
- Enlargement (scale 2:1) means drawing is twice the actual size
Steps for Scale Drawing:
- Identify the scale factor
- Multiply all actual dimensions by scale factor
- Construct figure using scaled measurements
- Mark the scale clearly on drawing
Real-world Applications: Maps, architectural plans, model buildings, engineering blueprints.
Exam Tip
Always mention and apply the scale factor. Questions often ask to find actual measurements from scaled drawings.
Common Mistake
Forgetting to apply scale factor or calculating it in the wrong direction (actual/drawing instead of drawing/actual).
Chapter Summary
Practical Geometry teaches precision construction methods essential for accurate geometric figures. This chapter covers:
- Tools and Techniques: Proper use of compass and straightedge for exact constructions
- Perpendiculars and Bisectors: Constructing perpendicular bisectors and perpendiculars from points
- Angles: Bisecting angles and copying angles without protractor
- Triangles: Constructing triangles using SSS, SAS, ASA, and RHS criteria
- Scale Drawings: Creating proportional representations using scale factors
Exam Focus: Step-by-step constructions with clear arc markings, accurate measurements, triangle construction from various criteria, and scale drawing applications.