Chapter focus
Geometric constructions use tools and properties to make accurate figures. Tilings repeat shapes across a surface with no gaps or overlaps.
These original explanations accompany the named Ganita Prakash Part II chapter. Follow the textbook for the full construction sequence and visual activities.
1. Construct a triangle from three sides
To construct a triangle with side lengths 4 cm, 5 cm and 6 cm:
- Draw a 6 cm segment as the base.
- Set the compass to 4 cm and draw an arc from one endpoint.
- Set it to 5 cm and draw an arc from the other endpoint.
- Mark an intersection of the arcs as the third vertex and join it to both endpoints.
The arcs intersect because 4 + 5 > 6, so the side lengths satisfy the triangle inequality. A ruler-only sketch is less reliable than preserving the lengths with a compass.
2. Make and check a tiling
A tiling covers a flat surface using repeated shapes without gaps or overlaps. Around any point where shapes meet, the angles must total 360°.
- Equilateral triangles meet 6 at a point: 6 × 60° = 360°.
- Squares meet 4 at a point: 4 × 90° = 360°.
- Regular hexagons meet 3 at a point: 3 × 120° = 360°.
Regular pentagons have interior angle 108°. Three leave a gap (324° total), while four overlap (432°), so identical regular pentagons alone do not tile the plane by meeting at one vertex.
3. Combine shapes and repeat a pattern
Different polygons may tile together if the angles meeting at each vertex sum to 360° and the edges fit without gaps. Check both the angle total and how the sides meet; a correct angle sum alone may not ensure a seamless repeating pattern.
Use tracing or cut-outs to test a design, then describe the repeating unit and the transformations—such as a slide, turn or reflection—that reproduce it.
Common errors to avoid
- Keep compass openings unchanged when transferring a required length.
- Check that constructed sides match all given measurements.
- For a tiling, test the angles around a vertex and check edge alignment.
- A pattern that fills one small patch may still fail to repeat without gaps.
Practice: construct and reason
1. Can 4 cm, 5 cm and 6 cm form a triangle?
Answer: Yes. The two shorter sides sum to 9 cm, which is greater than 6 cm.
2. How many squares meet around one point in a regular square tiling?
Answer: Four, because 4 × 90° = 360°.
3. Why do three regular hexagons fit around a point?
Answer: Each interior angle is 120°, and 3 × 120° = 360°.
4. Can four regular pentagons meet around a point without overlap?
Answer: No. Their angles total 4 × 108° = 432°, more than 360°.
5. What tools preserve the side lengths in an SSS triangle construction?
Answer: A ruler or straightedge for the base and a compass to transfer the other two lengths.
Chapter summary
Use a compass and straightedge to preserve lengths in geometric constructions. A tiling repeats shapes to cover a surface; check the angles around a vertex total 360° and verify that edges meet without gaps or overlaps.