Playing with Constructions Chapter Notes | Maths for Class 6 (Ganita Prakash) - New NCERT PDF Download

Introduction

In this chapter, we will explore the fascinating world of geometric constructions. Using tools like a ruler and a compass, we will learn how to draw various shapes, including circles, squares, rectangles, and more complex figures. By the end of this chapter, you will have a good understanding of how to construct these figures accurately and will be able to recreate them on your own.

Artwork

Understand how to draw freehand figures and then use geometric tools to replicate these figures more accurately.

Key Concepts

• Curves and Lines: Curves include any shape that can be drawn on paper, such as straight lines, circles, and other figures.
  
• Using a Compass: A compass is used to draw circles and arcs. By marking a point and adjusting the compass to a specific length, we can draw a circle with that point as the center.

Example

Mark a point 'P' on your notebook. 

Now, use a compass to mark points that are 4 cm away from P in different directions. 
Join these points to form a circle. Here, 'P' is the center, and the distance (4 cm) is the radius of the circle.

Squares and Rectangles

Understanding Rectangles

• Rectangle ABCD: Let's take a closer look at a rectangle named ABCD.
  
• Corners and Sides: The points A, B, C, and D are the corners of the rectangle. The lines AB, BC, CD, and DA are its sides. The angles at these corners are called ∠A, ∠B, ∠C, and ∠D.
• Opposite Sides: The blue sides AB and CD are called opposite sides because they are across from each other. Similarly, AD and BC form another pair of opposite sides.

Properties of Rectangles

• Opposite Sides: In a rectangle, opposite sides are equal in length.
• Angles: All angles in a rectangle are 90 degrees.

Naming Rectangles

• The rectangle in the picture is named ABCD, but it can also be called BCDA, CDAB, DABC, ADCB, DCBA, CBAD, and BADC.

Valid Naming

• A rectangle can be named using any combination of the corner labels, but not all combinations are allowed.
• For example, names like ABDC or ACBD are not valid.
• A valid name must list the corners in the order you travel around the rectangle, starting from any corner.

Understanding Squares

Corners and Sides: Just like in a rectangle, the corners and sides of a square are defined in the same way.

Properties of a Square: A square has two key properties:

• All sides are equal in length.
• All angles are 90 degrees.

Rotated Squares and Rectangles

When we rotate a square piece of paper, we want to see if it still remains a square. 

Let's check the properties:

• Are all the sides still equal? Yes, they are.
• Are all the angles still 90 degrees? Yes, they are.

Rotating a square does not change the lengths of its sides or the measures of its angles. Since the rotated figure still meets the properties of a square, it is still a square.

Similarly, when we rotate a rectangle, it remains a rectangle for the same reasons.

Constructing Squares and Rectangles

Understanding of Constructing Squares and Rectangles with the help of Example

How to Construct a Square with a Side Length of 6 cm?

Follow these steps to construct a square with each side measuring 6 cm:

Step 1: Create a square PQRS with a side length of 6 cm.

Step 2: At point P, draw a perpendicular line to PQ.

Step 3: Method 1: Using a ruler, mark point S on the perpendicular line such that PS = 6 cm.

Method 2: You can also use a compass to measure PS.

Step 4: Draw a perpendicular line to line segment PQ at point Q.

Step 5: If using a compass, the next point can be easily marked with it.

Step 6: Complete the square by ensuring all sides are equal.

An Exploration in Rectangles

A rectangle is a four-sided shape where opposite sides are equal in length, and all four angles are right angles (90 degrees). Imagine you have a piece of paper in the shape of a rectangle. The longer sides are usually called the length, and the shorter sides are called the width.

Understanding with Examples

Constructing a Rectangle (ABCD)

1. Step 1: Draw Rectangle ABCD

   • Draw a rectangle ABCD where side AB = 7 cm and side BC = 4 cm.
   • Label the corners of the rectangle as A, B, C, and D.

2. Step 2: Points X and Y

   • Imagine point X can move along side AD (from A to D).
   • Similarly, point Y can move along side BC (from B to C).
   • X can be on A or D, and Y can be on B or C.

3. Step 3: Measuring Distance XY

   • Place X and Y at different points on AD and BC.
   • Measure the distance between X and Y by drawing a line XY.
   • Observe how this distance changes depending on the positions of X and Y.

Key Observation: The shortest distance between X and Y will always be along the line parallel to AB.

Exploring Diagonals of Rectangles and Squares

When we draw a rectangle and connect its opposite corners with lines, we create two lines called the diagonals. These diagonals are important because they help us understand some special properties of the rectangle.

1. Understanding Diagonals: In a rectangle, when we join the opposite corners, we get two diagonals. These diagonals are equal in length.
2. Exploring Angles: When we draw one of the diagonals, it splits the angles at the corners of the rectangle into two smaller angles. For example, the diagonal divides angle R into two smaller angles, which we call g and h. Similarly, it divides angle P into c and d. We can measure these angles to see if g and h are equal and if c and d are equal.
3. Making Predictions: Before measuring the angles, we can make predictions about whether the angles will be equal or not. After measuring, we can see if our predictions were right.
4. Special Case of a Square: When all four sides of the rectangle are equal, we have a square. In this case, the diagonals also have special properties. It’s interesting to see how the angles and sides behave differently in a square compared to a rectangle.
5. General Observations: As we explore different rectangles and squares, we can notice patterns about the angles and sides. For example, we might find that certain angles are always equal, or that the diagonals are always the same length.
6. Ensuring Accuracy: To make sure our observations are correct, we can check them with different rectangles and squares. This helps us confirm that the rules we see are always true, no matter what rectangle or square we have.
7. Construction Challenges: Sometimes, we might be asked to create a rectangle with specific angles or side lengths. This can be a fun challenge!

Understanding with Examples

Example 1: Constructing a Rectangle with Specific Angles
Let’s say we want to build a rectangle where one of the corners is split into angles of 60° and 30°.

1. Step 1: We can start by drawing a base line, which we’ll call AB. This line can be any length we choose.
   
2. Step 2: Next, we need to find point D. To do this, we draw a line from point A that goes straight up, making a right angle with AB. This helps us find point D, which has to be somewhere on this line.
   
3. Step 3: Now, we need to create the angles at point A. We know one angle has to be 60°, so we can draw that angle. The other angle will be whatever is left to make a total of 90° because all corners in a rectangle are right angles.
   
4. Step 4: Once we have point D, we can finish the rectangle by connecting the dots. We can either draw perpendicular lines from points D and B to find the last corner, or we can use the fact that opposite sides of a rectangle are equal to find the missing point.
   
5. Step 5: Finally, we check to make sure our rectangle looks right. We can adjust if needed, but this is the basic idea of how to construct a rectangle with specific angles!
   

Example 2: Constructing a Rectangle with Given Side and Diagonal
Sometimes, we need to create a rectangle when we know the length of one side and the diagonal. Let’s say one side is 5 cm long and the diagonal is 7 cm long.

• Step 1: We start by drawing the base of the rectangle, called CD, which is 5 cm long.
  
• Step 2: Next, we need to draw a line straight up from point C. This line will help us find point B, which needs to be somewhere on this line.
  
• Step 3: Now, we need to find point B. We know that B has to be 7 cm away from point D. Instead of guessing where B is, we can draw a circle with a radius of 7 cm around point D. The point where this circle intersects the line going up from point C will be point B.
  
• Step 4: Once we have points C, D, and B, we can complete the rectangle by finding the last corner, point A. We do this by drawing lines from points D and B that go straight out to meet each other.
  

Points Equidistant from Two Given Points

When you have two points, you can find all the points that are the same distance from both of them. This set of points will create a line that goes right in the middle of the two points.

Example: Let’s say we have two points: B at (1, 2) and C at (5, 2). To find all the points that are the same distance from both B and C:

• Draw a line between B and C: Connect the two points with a straight line.
• Use a compass to draw arcs: From point B, draw two arcs above and below the line BC. Then do the same from point C.
  
• Find the intersection points: The arcs will intersect at two points.
  
• Draw the perpendicular bisector: Connect the two intersection points with a straight line. This line represents all the points that are equidistant from B and C.
  

In this example, the perpendicular bisector would be the line that runs right down the middle between points B and C, showing all the points that are the same distance from both.

Key Words

• All the points of a circle are at the same distance from its centre. This distance is called the radius of the circle.
• A compass can be used to construct circles and their parts.
• A rough diagram can be useful in planning how to construct a given figure.
• A rectangle can be constructed given the lengths of its sides or that of one of its sides and a diagonal.