Practical Geometry
1. Geometry Tools and Their Uses
| Tool | Use |
|---|---|
| Ruler | Drawing straight lines, measuring lengths |
| Compass | Drawing circles, arcs, copying lengths |
| Protractor | Measuring and drawing angles |
| Divider | Comparing lengths, marking equal segments |
| Set squares | Drawing perpendicular and parallel lines, constructing specific angles |
2. Constructing a Circle
To draw a circle of given radius (say 4 cm):
Step 1: Mark center O.
Step 2: Open compass to 4 cm using a ruler.
Step 3: Place the pointed end at O and rotate the pencil end to draw the circle.
Step 4: Label the radius OA.
Key terms: Center, radius (distance from center to circle), diameter (2 x radius), chord (line joining two points on the circle), circumference (the boundary).
Common Mistake: Changing the compass opening while drawing the circle. Keep it fixed.
3. Constructing a Line Segment
To construct a line segment of length 6.5 cm:
Step 1: Mark point A on paper.
Step 2: Place ruler with 0 at A.
Step 3: Mark point B at 6.5 cm.
Step 4: Join A and B.
To copy a line segment using a compass:
Step 1: Measure the given segment AB with the compass.
Step 2: Draw a line, mark point P.
Step 3: With the same compass opening, draw an arc from P to cut the line at Q.
PQ = AB.
4. Perpendicular Bisector
A line that divides a segment into two equal parts at 90.
Construction of perpendicular bisector of AB:
Step 1: With center A, radius more than half of AB, draw arcs above and below AB.
Step 2: With center B and the same radius, draw arcs intersecting the first arcs at P and Q.
Step 3: Join P and Q. Line PQ is the perpendicular bisector.
Let PQ meet AB at M. Then AM = MB and PQ is perpendicular to AB.
Exam Focus (3 marks): 'Draw a segment of 7 cm. Construct its perpendicular bisector and verify that it divides the segment equally.'
Follow the steps above. Measure AM and MB. Both should be 3.5 cm.
5. Angle Bisector
A ray that divides an angle into two equal parts.
Construction of angle bisector of angle ABC:
Step 1: With vertex B as center, draw an arc cutting BA at P and BC at Q.
Step 2: With center P, draw an arc inside the angle.
Step 3: With center Q and same radius, draw an arc intersecting the previous arc at R.
Step 4: Draw ray BR. BR bisects angle ABC.
Common Mistake: Using different radii for the two interior arcs. Both interior arcs must have the same radius.
6. Constructing Standard Angles
Constructing 60
Step 1: Draw ray AB.
Step 2: With A as center, draw an arc cutting AB at P.
Step 3: With P as center and same radius, draw an arc cutting the first arc at Q.
Step 4: Draw ray AQ. Angle QAB = 60.
Constructing 120
Follow the 60 construction. With Q as center and same radius, draw another arc on the first arc at R. Angle RAB = 120.
Constructing 30
Construct 60, then bisect it (construct the angle bisector of 60). Each part = 30.
Constructing 90
Method 1: Draw a perpendicular to a line.
Method 2: Construct 60 and 120 together. Bisect the 120 angle to get 60. Or: draw the perpendicular bisector of a segment.
Exam Focus (4 marks): 'Construct angle of 60 and then construct its angle bisector to get 30. Label all construction lines.'
Construct angle CAB = 60. bisect it with ray AD. Angle DAB = 30.
7. Comparison Table: Constructions
| Construction | Tools Needed | Key Property |
|---|---|---|
| Circle | Compass, ruler | All points at fixed distance from center |
| Line segment | Ruler | Definite length |
| Perpendicular bisector | Compass, ruler | Divides segment equally at 90 |
| Angle bisector | Compass, ruler | Divides angle into two equal parts |
| 60 angle | Compass, ruler | Equilateral triangle property |
8. Self-Test
- Draw a circle of radius 3.5 cm. Label the center, a radius, and a diameter.
- Draw a line segment of length 5.8 cm.
- Construct the perpendicular bisector of a segment of length 7.4 cm.
- Construct an angle of 60 and then construct 30 using an angle bisector.
- Construct an angle of 120 using a compass only (no protractor).
- What is the difference between drawing a line segment and constructing a copy using a compass?
- Construct an angle of 90 using a ruler and compass.
9. Answers to Self-Test
- Draw using compass with radius 3.5 cm. Label O as center, OA as radius, BC as diameter passing through O.
- Draw using ruler: mark 0 and 5.8 cm points, join them.
- Follow perpendicular bisector construction steps. Verify: measure from intersection point to endpoints (3.7 cm each).
- Construct 60 first using compass arcs, then construct angle bisector to get 30.
- Construct 60, then extend to the second arc for 120.
- Ruler method marks endpoints directly; compass method transfers length without relying on the ruler scale.
- Construct perpendicular line: draw a line and construct perpendicular bisector of a segment on it, or construct 60 and adjacent 120, then bisect.
