Practical Geometry Application | Class 6 Maths NCERT

What is Practical Geometry?

Practical Geometry is the branch of mathematics where we learn to draw and construct geometric figures accurately. But what is the difference between drawing and constructing? A 'drawing' or 'sketch' is a rough, freehand picture that gives you an idea of the shape. A 'construction', on the other hand, is a very precise figure made using specific tools. In Class 6, our main tools are an unmarked ruler (also called a straightedge) and a pair of compasses. The ruler helps us draw straight lines, and the compasses help us draw arcs and circles of a fixed radius. The real magic happens when we combine these tools to create complex figures like perpendicular bisectors, angle bisectors, and specific angles (like 30°, 45°, 60°, 90°, 120°) just by following logical steps. This process teaches us the underlying properties of shapes.

How to Construct a Perpendicular Bisector

1. Step 1: Draw the Line Segment — Use your ruler to draw a line segment of any length. Let's call the endpoints A and B. For example, draw a line segment AB of length 8 cm.
2. Step 2: Set Your Compass — Place the metal point of your compass at point A. Open the compass so that the pencil end is more than half the length of AB. This is important! If it's less than half, the arcs won't intersect.
3. Step 3: Draw Arcs from Point A — With the compass set, draw a large arc above the line segment AB and another large arc below the line segment AB.
4. Step 4: Draw Arcs from Point B — Without changing the width of the compass, move the metal point to B. Now, draw two more arcs, one above and one below AB, so that they intersect the first two arcs. Let's call the intersection points P (above AB) and Q (below AB).
5. Step 5: Join the Intersection Points — Use your ruler to draw a straight line connecting points P and Q. This line PQ is the perpendicular bisector of the line segment AB. It cuts AB at a perfect 90° angle and divides AB into two equal halves.

Worked Example: Constructing a 60° Angle

• Constructing a 60° angle is one of the most fundamental constructions. Here’s how you do it using only a ruler and compasses. Goal: Construct an angle of 60°. Step 1: Draw a Ray. Use your ruler to draw a straight ray. Let's name it OA, where O is the starting point. Step 2: Draw an Arc. Place the metal tip of your compass on point O. Open the compass to any convenient radius (e.g., 3 cm) and draw a large arc that cuts the ray OA at a point. Let's call this point P. Step 3: Draw the Intersecting Arc. Without changing the compass width, move the metal tip to point P. Draw another arc that intersects the first arc at a new point. Let's call this point Q. Step 4: Join to Form the Angle. Use your ruler to draw a ray starting from point O and passing through point Q. Let's call this ray OB. Result: The angle you have just formed, ∠AOB, is exactly 60°. This works because you have created the corner of an equilateral triangle, where all angles are 60°.

Key Tips for Accurate Constructions

• Use a Sharp Pencil: A thick pencil line can make your measurements and intersections inaccurate. Always use a well-sharpened pencil for both your compass and your ruler.
• Keep Your Compass Tight: Make sure the hinge and pencil holder of your compass are tight. A loose compass will change its width while you are drawing, ruining your construction.
• Draw Light Construction Lines: The arcs and initial lines you draw to find your points are called construction lines. Draw them lightly so they don't get confused with the final, darker lines of your required figure.
• Do Not Erase Your Arcs: Teachers and examiners need to see your construction arcs to understand your method. Erasing them makes it look like you guessed or used a protractor.

Practice Questions with Solutions

• Q: Draw a circle with a radius of 5.2 cm. A: Step 1: Use a ruler to measure 5.2 cm. Step 2: Open your compasses to this width, placing the metal tip on the '0' mark and the pencil tip on the '5.2 cm' mark of the ruler. Step 3: Mark a center point 'O' on your paper. Step 4: Place the metal tip of the compass on point O and rotate the pencil end 360 degrees to draw the circle. Final answer: You have constructed a circle with center O and radius 5.2 cm.
• Q: Draw a line segment of length 9 cm and construct its perpendicular bisector. A: Step 1: Use a ruler to draw a line segment AB of length 9 cm. Step 2: Place the compass point at A and draw arcs above and below AB, with a radius more than half of 9 cm (e.g., 5 cm). Step 3: Without changing the compass width, place the compass point at B and draw arcs to intersect the previous arcs at points P and Q. Step 4: Join P and Q with a ruler. The line PQ is the perpendicular bisector of AB. Final answer: The line PQ intersects AB at its midpoint at a 90-degree angle.
• Q: Construct an angle of 120° using a ruler and compasses. A: Step 1: Draw a ray OA. Step 2: With O as the center, draw an arc that cuts OA at point P. Step 3: With P as the center and the same radius, cut the arc at Q. (This creates a 60° angle ∠QOA). Step 4: With Q as the center and the same radius, cut the arc again at a new point R. Step 5: Join O to R. The angle ∠ROA is the required 120° angle. Final answer: The constructed angle ∠ROA measures 120° (60° + 60°).
• Q: Construct a copy of a given angle ∠ABC = 45° without using a protractor. A: Step 1: Draw the given angle ∠ABC = 45° (you can use a protractor for this initial drawing). Step 2: Now, draw a new ray YZ. This will be one arm of your new angle. Step 3: On the original angle, place the compass point at B and draw an arc cutting BA at D and BC at E. Step 4: Without changing the compass width, place the compass point at Y and draw a similar arc cutting YZ at a point S. Step 5: Go back to the original angle. Measure the distance between D and E with your compass. Step 6: Keeping this width, place the compass point at S on the new drawing and draw an arc to intersect the arc from Step 4. Call this point T. Step 7: Join Y and T to form the ray YX. Final answer: The new angle ∠XYZ is an exact copy of ∠ABC and measures 45°.

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