# Playing with Constructions

## Overview

This chapter introduces geometric constructions using a compass and a ruler. Students explore shapes like circles, squares, and rectangles through activities and step-by-step construction tasks. It helps build spatial reasoning and an understanding of shape properties such as angles, sides, and diagonals.

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## Key Topics Covered

### 1. Using a Compass – Understanding Circles

- A compass helps draw shapes by fixing the tip and rotating the pencil.
- **Circle**: All points at the same distance (radius) from a fixed point (centre).
- Activity: Draw a circle with a 4 cm radius using a compass.
- Terms introduced: *radius*, *centre*.

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### 2. Artwork with Constructions

- Try drawing artistic figures using a compass:
- **A Person** – curved features drawn with trial placements.
- **Wavy Wave** – built using half-circles along a line (e.g., 8 cm).
- **Eyes** – symmetrical arcs with compass; accuracy matters.

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### 3. Squares and Rectangles

#### Properties:

- **Rectangle**:
- Opposite sides are equal (R1).
- All angles are 90° (R2).
- **Square**:
- All sides are equal (S1).
- All angles are 90° (S2).
- Names must follow corner order (e.g., ABCD, BCDA valid; ABDC not valid).
- Rotated squares/rectangles maintain their properties.

#### Activity:

- Draw squares/rectangles on dot paper.
- Identify equal sides and right angles using visual judgement.

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### 4. Constructing Squares and Rectangles

- **Square of side 6 cm**: Constructed using ruler and compass with perpendiculars.
- **Rectangle Construction Examples**:
- 4 cm × 6 cm
- 2 cm × 10 cm
- Discussed: Possibility of 4-sided figure with all right angles but unequal opposite sides.

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### 5. Exploring Rectangles

- Points on opposite sides (like X and Y on AD and BC) create line segments (XY).
- Activity: Move points and measure XY; record in a table.
- Observe patterns, especially when X and Y are equidistant from ends.
- Understand the geometry of *ABYX* figures.

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### 6. Breaking Rectangles into Squares

- Construct rectangles from 2 or 3 identical squares.
- Visualise before constructing.
- Use compass to transfer side lengths without ruler.

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### 7. Constructing Complex Figures

- **Square in Rectangle**: 8 cm × 4 cm rectangle with centred square.
- **Falling Squares**: Overlapping squares of different sizes.
- **Shading**: Create symmetric pattern using multiple small squares.
- **Square with a Hole**: Draw a square with a circle cut out in the centre.
- **Square with Curves**: Uniform arcs on each side using compass.

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### 8. Diagonals in Rectangles and Squares

- **Diagonals**: Lines joining opposite corners (e.g., PR and QS).
- Activity: Measure diagonals and angles they form at corners.
- Predict which angles are equal, verify by construction.
- Observe symmetry and draw conclusions for squares vs rectangles.

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### 9. Constructing with Given Angles or Diagonals

- **Examples**:
- Diagonal divides angles into 60° and 30°.
- Rectangle with side 5 cm and diagonal 7 cm.
- Use compass to find points at fixed distance (construct arcs).

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### 10. Equidistant Point Construction

- Construct figure like a house with all sides 5 cm.
- **Point A** must be 5 cm from both B and C.
- Use intersecting arcs (from B and C) to locate A.
- Extend idea to other shapes and repeat for larger sizes or different figures.

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## New Terms and Simple Definitions

| Term | Simple Definition |
|--------------|---------------------------------------------------------------------|
| Compass | A tool to draw circles and arcs by fixing one point and rotating. |
| Radius | Distance from the centre of a circle to any point on its edge. |
| Centre | The fixed point in the middle of a circle. |
| Diagonal | A line joining opposite corners of a rectangle or square. |
| Perpendicular| Two lines that meet at a right angle (90°). |
| Arc | A curved part of a circle. |
| Square | A shape with 4 equal sides and all angles 90°. |
| Rectangle | A shape with opposite sides equal and all angles 90°. |
| Angle | The measure of turn between two lines from a common point. |
| Dot grid | Paper with evenly spaced dots used to help draw geometric shapes. |

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## Practice Problems

### Easy (3)

1. **Draw a circle** of radius 3 cm using a compass.
**Answer**: Fix compass at a point, adjust to 3 cm, rotate to complete the circle.

2. **Identify the radius** in a circle if the distance from centre to edge is 5 cm.
**Answer**: Radius = 5 cm.

3. **What shape has four equal sides and all right angles?**
**Answer**: Square.

### Medium (2)

4. **Construct a rectangle** of length 6 cm and breadth 4 cm. Verify opposite sides.
**Answer**: Use ruler and compass, draw perpendiculars, check sides with ruler.

5. **Draw two intersecting arcs** to find a point equidistant from points A and B (5 cm apart).
**Answer**: Draw circles from A and B with same radius; intersection gives the required point.

### Difficult (3)

6. **Construct a square inside a rectangle** of 8 cm × 4 cm with same centre.
**Answer**: Find midpoints of opposite sides, draw perpendicular bisectors, use for square.

7. **Draw a rectangle with a diagonal dividing angle into 60° and 30°**.
**Answer**: Use protractor to set angle, construct lines, ensure all sides and angles satisfy rectangle properties.

8. **Measure and compare diagonals** of a square drawn on dot grid.
**Answer**: Use ruler or count dots; both diagonals are equal.

### Very Difficult (2)

9. **Construct a rectangle given one side (4 cm) and diagonal (7 cm)**.
**Answer**: Draw base, construct perpendicular, draw arc of 7 cm radius from one end; where arc meets perpendicular is the corner.

10. **Can you construct a quadrilateral with all sides equal but no right angle?**
**Answer**: Yes, it’s a rhombus. Construct with compass: same-length sides, but angles not 90°.

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