Constructions
Lesson Objectives
- Understand basic terms in geometric constructions.
- Construct angles, perpendiculars, and bisectors using a compass and ruler.
- Replicate geometric shapes and segments accurately without measurement.
- Solve construction-based problems geometrically.
Lesson Introduction
Geometric construction is a classical method of drawing shapes, angles, and lines using only a compass and a straightedge. It is a foundational concept in geometry and useful in design, architecture, and technical drawing. This lesson explores how to perform precise constructions step-by-step.
Core Lesson Content
Key Tools: Compass, ruler (straightedge), pencil.
Important Terms:
- Bisector: A line that divides an angle or segment into two equal parts.
- Perpendicular: A line that intersects another at a 90-degree angle.
- Arc: A part of a circle drawn by a compass.
Worked Examples
Example 1: Construct an angle of 60° at a point.
Steps:
- Draw a baseline and mark a point A on it.
- With the compass on A, draw an arc that cuts the baseline at B.
- Without changing the compass width, place the compass at B and draw an arc cutting the previous arc at C.
- Join A and C. ∠CAB = 60°.
Steps:
- Draw a baseline and mark a point A on it.
- With the compass on A, draw an arc that cuts the baseline at B.
- Without changing the compass width, place the compass at B and draw an arc cutting the previous arc at C.
- Join A and C. ∠CAB = 60°.
Example 2: Bisect a given angle.
Steps:
- Draw angle ∠XYZ.
- With the compass on Y, draw an arc to cut both arms at A and B.
- From A and B, draw arcs to intersect at C.
- Draw YC. This bisects the angle.
Steps:
- Draw angle ∠XYZ.
- With the compass on Y, draw an arc to cut both arms at A and B.
- From A and B, draw arcs to intersect at C.
- Draw YC. This bisects the angle.
Example 3: Construct a perpendicular from a point to a line.
Steps:
- Draw a line and a point P above it.
- From P, draw arcs to intersect the line at A and B.
- With equal radius, draw arcs from A and B to meet at C.
- Connect P to C. Line PC is perpendicular.
Steps:
- Draw a line and a point P above it.
- From P, draw arcs to intersect the line at A and B.
- With equal radius, draw arcs from A and B to meet at C.
- Connect P to C. Line PC is perpendicular.
Example 4: Construct a perpendicular bisector of a line segment.
Steps:
- Draw segment AB.
- With radius > half AB, draw arcs above and below from A and B.
- Join the intersection points. This line bisects AB perpendicularly.
Steps:
- Draw segment AB.
- With radius > half AB, draw arcs above and below from A and B.
- Join the intersection points. This line bisects AB perpendicularly.
Example 5: Construct an angle of 90°.
- First construct a 60° and 120° as in Example 1.
- Bisect angle between them to get 90°.
- First construct a 60° and 120° as in Example 1.
- Bisect angle between them to get 90°.
Example 6: Construct a triangle given all sides (SSS).
- Draw the base AB = 5 cm.
- From A, draw arc 6 cm; from B, draw arc 4 cm to meet at C.
- Connect AC and BC.
- Draw the base AB = 5 cm.
- From A, draw arc 6 cm; from B, draw arc 4 cm to meet at C.
- Connect AC and BC.
Example 7: Construct a triangle given ASA.
- Draw base AB = 6 cm.
- At A and B, construct angles 40° and 60°.
- Extend rays to meet at C.
- Draw base AB = 6 cm.
- At A and B, construct angles 40° and 60°.
- Extend rays to meet at C.
Example 8: Construct a triangle given RHS (right angle, hypotenuse, side).
- Draw base AC = 5 cm.
- At C, construct 90° and mark hypotenuse 7 cm from A.
- Connect the triangle.
- Draw base AC = 5 cm.
- At C, construct 90° and mark hypotenuse 7 cm from A.
- Connect the triangle.
Example 9: Construct a square of side 4 cm.
- Draw base AB = 4 cm.
- Construct 90° at A and B.
- Mark 4 cm from each point. Connect to complete square.
- Draw base AB = 4 cm.
- Construct 90° at A and B.
- Mark 4 cm from each point. Connect to complete square.
Example 10: Construct a hexagon of side 3 cm.
- Draw a circle of radius 3 cm.
- Place compass on the circle's edge, step around to mark 6 arcs.
- Connect points to form hexagon.
- Draw a circle of radius 3 cm.
- Place compass on the circle's edge, step around to mark 6 arcs.
- Connect points to form hexagon.
Exercises
- Construct a 45° angle using only a compass and ruler.
- [NABTEC] Bisect an angle of 120°. [Past Question]
- Construct a triangle with sides 5 cm, 6 cm, and 7 cm.
- Construct a perpendicular to a given line from a point on the line.
- [WAEC] Construct a triangle with base 7 cm and angles 60° and 50°. [Past Question]
- Construct a square of side 6 cm.
- [NECO] Construct an equilateral triangle with side 5 cm. [Past Question]
- Construct the perpendicular bisector of a 10 cm line segment.
- Construct a triangle given angle A = 45°, B = 60°, and AB = 6 cm.
- [JAMB] Using only compass and ruler, construct an angle of 30°. [Past Question]
Conclusion/Recap
This lesson introduced geometric construction techniques using compass and ruler. Students learned to construct angles, perpendiculars, triangles, and polygons accurately. Mastering these skills enhances spatial reasoning and geometric understanding. In the next lesson, we will explore Loci and its applications in geometry.
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