NCERT Solutions for Class 6 Maths Chapter 14 — Practical Geometry

Key Concepts

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Exercise 14.1 — Constructing Line Segments

Question 1

Q: Draw a line segment of length 7.3 cm using a ruler.

The line segment AB = 7.3 cm.

Verification: Place the ruler along AB and check that the distance from A to B is exactly 7.3 cm.

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Question 2

Q: Construct a line segment of length 5.6 cm using a ruler and compass.

The line segment AB = 5.6 cm.

Advantage of using a compass: The compass ensures accuracy because the length is set once and transferred without measurement error. A ruler alone may have parallax error.

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Question 3

Q: Construct AB of length 7.8 cm. From this, cut off AC of length 3.4 cm. Measure BC.

Verification: Measure BC with the ruler. It should read 4.4 cm.

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Question 4

Q: Given AB of length 3.9 cm, construct PQ such that PQ = AB.

Step 1: Observe the given line segment AB = 3.9 cm. Step 2: Draw a line l and mark a point P on it. Step 3: Place the compass tip on point A and the pencil tip on point B. The compass now captures the length AB. Step 4: Without changing the compass opening, place the compass tip on point P. Step 5: Draw an arc cutting line l. Mark the intersection as point Q.

Now, PQ = AB = 3.9 cm.

Key principle: A compass transfers an exact length without needing to know or read the numerical measurement.

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Question 5

Q: Given two line segments AB and CD, verify that AB = CD or AB > CD or AB < CD using a compass.

Step 1: Place the compass tip on A and the pencil tip on B. The compass now holds the length AB. Step 2: Without changing the compass opening, place the compass tip on C. Step 3: Observe the position of the pencil tip relative to point D:

• If the pencil tip falls exactly on D, then AB = CD.
• If the pencil tip falls beyond D (past D on the CD side), then AB > CD.
• If the pencil tip falls short of D, then AB < CD.

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Exercise 14.2 — Constructing a Copy of a Line Segment

Question 1

Q: Draw any line segment AB. Mark any point C on it. Construct a copy of AC using a compass.

PQ = AC. We have constructed a copy of AC.

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Question 2

Q: Given a line segment AB, construct a line segment CD such that CD = 2 times AB.

Step 1: Draw a line l and mark a point C on it. Step 2: Place the compass tip on A and pencil on B to capture the length AB. Step 3: Without changing the compass, place the tip on C and draw an arc cutting line l. Call this point E. So CE = AB. Step 4: Now, still using the same compass opening (length AB), place the tip on E and draw another arc cutting line l. Call this point D.

Now, CD = CE + ED = AB + AB = 2AB.

Verification: Measure CD with a ruler. It should be twice the length of AB.

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Question 3

Q: Given a line segment AB = 5.6 cm, construct a line segment of length 2.8 cm (half of AB).

Step 1: Draw line segment AB = 5.6 cm. Step 2: Construct the perpendicular bisector of AB:

Step 3: The perpendicular bisector cuts AB at its midpoint M. Step 4: AM = MB = 5.6 / 2 = 2.8 cm. Step 5: To construct a separate copy of AM = 2.8 cm, use the compass method: place tip on A, pencil on M, then transfer this length to a new line.

• With compass radius greater than half of AB, draw arcs from A above and below AB.
• With the same radius, draw arcs from B above and below AB.
• The arcs from A and B intersect at two points. Join these points.

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Question 4

Q: Construct a line segment of length 3.4 cm and then construct a copy of it using a compass. Verify that both are equal.

Step 2: To construct a copy:

Step 3: Verify CD = AB:

• Draw a line l, mark point A.
• Set compass to 3.4 cm on the ruler.
• From A, draw an arc cutting line l at B.
• Draw another line m and mark point C.
• Place compass tip on A and pencil on B (compass captures AB = 3.4 cm).
• Without changing compass, place tip on C and draw an arc cutting line m at D.
• Place compass on C and pencil on D (captures CD).
• Without changing, place compass tip on A.
• If the pencil tip falls exactly on B, then CD = AB.

Verification: The pencil tip falls exactly on B, confirming CD = AB = 3.4 cm.

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Exercise 14.3 — Constructing Circles

Question 1

Q: Draw a circle of radius 3.2 cm.

The resulting circle has centre O and radius 3.2 cm.

Key points:

• The centre is the fixed point where the compass tip is placed.
• The radius is the distance from the centre to any point on the circle.
• The diameter = 2 x radius = 2 x 3.2 = 6.4 cm.

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Question 2

Q: With the same centre O, draw two circles of radii 3 cm and 5 cm.

The two circles are concentric circles — they share the same centre O but have different radii.

Observations:

• Circle 1 (inner): radius = 3 cm, diameter = 6 cm.
• Circle 2 (outer): radius = 5 cm, diameter = 10 cm.
• The region between the two circles is called a ring or annulus.

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Question 3

Q: Draw a circle of radius 4 cm. Draw its diameter. Also draw a chord that is not a diameter.

Step 3: To draw a chord that is not a diameter:

• Draw a line through centre O that cuts the circle at two points A and B.
• AB is a diameter. AB = 2 x 4 = 8 cm.
• Mark two points P and Q on the circle such that the line PQ does NOT pass through the centre O.
• PQ is a chord but not a diameter.

Difference between diameter and chord:

• A chord is any line segment whose both endpoints lie on the circle.
• A diameter is a special chord that passes through the centre.
• The diameter is the longest chord of a circle.

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Question 4

Q: Draw a circle of radius 2.5 cm and mark: (a) its centre (b) a radius (c) a diameter (d) a chord that is not a diameter

Summary:

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Exercise 14.4 — Constructing Perpendiculars and Perpendicular Bisectors

Question 1

Q: Draw a line segment AB of length 6.4 cm and construct its perpendicular bisector.

Verification:

• Measure AM and MB. Both should be 3.2 cm (half of 6.4 cm).
• Measure angle PMA with a protractor. It should be 90 degrees.

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Question 2

Q: Draw a line l and a point P on it. Construct a perpendicular to l through P using a compass.

Line PQ is perpendicular to line l at point P.

Verification: Measure angle APQ (or BPQ) with a protractor. It should be 90 degrees.

Key insight: This method works because we are essentially constructing the perpendicular bisector of AB, where A and B are equidistant from P on line l. Since P is the midpoint of AB, the perpendicular bisector passes through P and is perpendicular to line l.

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Question 3

Q: Draw a line l and a point P not on it. Construct a perpendicular from P to l using a compass.

Verification: Measure angle PMA (or PMB). It should be 90 degrees.

Note: PQ is the shortest distance from P to line l. The foot of the perpendicular M is the closest point on l to P.

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Question 4

Q: Draw a line segment AB = 5.8 cm. Construct the perpendicular bisector of AB. Verify that it bisects AB.

Verification:

• Measure AM: It should be 2.9 cm.
• Measure MB: It should be 2.9 cm.
• AM + MB = 2.9 + 2.9 = 5.8 cm = AB. (Verified)
• Measure angle AMP with a protractor. It should be 90 degrees. (Verified)

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Question 5

Q: Draw a line l and a point P on it. Construct a perpendicular to l at P. Also construct a line parallel to l through a point Q above l.

Part B: Parallel line through Q Step 1: Mark point Q above line l (not on l). Step 2: From Q, draw a perpendicular to line l using the method above. Let this perpendicular meet l at M. Step 3: Now, from Q, construct a perpendicular to QM (the perpendicular we just drew). Step 4: This new line through Q is perpendicular to QM and therefore parallel to l.

Principle: If two lines are both perpendicular to the same line, they are parallel to each other.

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Exercise 14.5 — Constructing Angles

Question 1

Q: Construct an angle of 60 degrees using a compass.

Angle AOC = 60 degrees.

Why this works: In triangle OBC, OB = OC (arcs of same radius from O) and BC = OB (arc drawn with same radius from B). So OBC is an equilateral triangle, and all angles of an equilateral triangle are 60 degrees.

Verification: Measure angle AOC with a protractor. It should be 60 degrees.

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Question 2

Q: Construct an angle of 30 degrees using a compass.

Step 1: Construct angle AOC = 60 degrees using the method from Question 1. Step 2: With B as centre and a convenient radius, draw an arc cutting OB and OC at points D and E respectively. Step 3: With D as centre and a radius greater than half of DE, draw an arc. Step 4: With E as centre and the same radius, draw another arc intersecting the previous arc at point F. Step 5: Join O to F with a ray OF.

Angle AOF = 30 degrees (half of 60 degrees).

Verification: Measure angle AOF with a protractor. It should be 30 degrees.

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Question 3

Q: Construct an angle of 90 degrees using a compass.

Method 2: Angle bisection method Step 1: Draw a line l and mark point O on it. Step 2: Construct angle AOC = 60 degrees at O on line l. Step 3: Construct angle COD = 60 degrees adjacent to angle AOC. Now angle AOD = 120 degrees. Step 4: Bisect angle AOD to get angle AOE = 60 degrees. Step 5: Bisect angle AOE to get angle AOF = 30 degrees.

Wait — let us use a simpler method:

Method 3: Direct construction Step 1: Draw a line l and mark point O on it. Step 2: With O as centre, draw a semicircle (arc) cutting line l at A and B (on opposite sides of O). Step 3: With A as centre and radius AO, draw an arc cutting the semicircle at C. Step 4: With B as centre and radius BO, draw an arc cutting the semicircle at D (other than A). Step 5: Join O to C. Angle AOC = 60 degrees. Step 6: The arc between C and D subtends angle COD = 60 degrees at O. Step 7: Bisect angle COD to get 30 degrees. Then angle AOB = angle AOC + angle COD = 120 degrees, and the supplementary angle is 60 degrees.

Simplest Method: Step 1: Construct angle of 60 degrees at point O on a line. Step 2: Construct another 60-degree angle adjacent to it (total = 120 degrees). Step 3: Bisect the 120-degree angle. The result is 60 degrees. Then the angle on the other side = 180 - 60 = 120. Wait, let me restate:

Correct approach: Step 1: Draw a ray OA. Step 2: Construct angle AOB = 60 degrees. Step 3: On the other side of OA, construct angle AOC = 60 degrees. Step 4: Now angle BOC = 60 + 60 = 120 degrees. Bisect angle BOC. Step 5: The bisector gives an angle of 60 degrees from OA. The remaining angle = 180 - 60 = 120. Hmm.

Best method — perpendicular: Step 1: Draw a line l and mark point O on it. Step 2: With O as centre, draw arcs cutting l at A and B on either side of O, with OA = OB. Step 3: With A as centre and radius greater than OA, draw an arc above. Step 4: With B as centre and the same radius, draw an arc intersecting the previous arc at C. Step 5: Join OC. Angle AOC = 90 degrees.

Verification: Measure angle AOC with a protractor. It should be 90 degrees.

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Question 4

Q: Construct an angle of 120 degrees using a compass.

Angle AOR = 120 degrees.

Why this works: Each arc of radius OP subtends 60 degrees at O. Two successive arcs give 60 + 60 = 120 degrees.

Verification: Measure angle AOR with a protractor. It should be 120 degrees.

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Question 5

Q: Construct an angle of 150 degrees using a compass.

Method: We know that 150 degrees = 180 degrees - 30 degrees. So, 150 degrees = 120 degrees + 30 degrees.

Step 1: Draw a ray OA. Step 2: Construct angle AOB = 120 degrees (as described above). Step 3: Construct angle BOC = 30 degrees (by bisecting a 60-degree angle) adjacent to angle AOB, on the opposite side. Step 4: Angle AOC = 120 + 30 = 150 degrees.

Alternative Method: Step 1: Draw a line l and mark point O on it. Step 2: Construct a 60-degree angle on one side. Step 3: The supplementary angle on the other side = 180 - 60 = 120 degrees. Step 4: Bisect the 60-degree angle to get 30 degrees. Step 5: Add 120 + 30 = 150 degrees.

Verification: Measure the constructed angle with a protractor. It should be 150 degrees.

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Question 6

Q: Construct an angle of 45 degrees using a compass.

Step 1: Draw a line l and mark point O on it. Step 2: Construct a perpendicular (90-degree angle) at O using the compass method from Exercise 14.4. Step 3: Now bisect the 90-degree angle:

Step 4: Ray OR bisects the 90-degree angle, giving angle AOR = 45 degrees.

• With O as centre, draw an arc cutting both arms of the 90-degree angle at points P and Q.
• With P as centre and radius greater than half PQ, draw an arc.
• With Q as centre and the same radius, draw an arc intersecting the previous arc at R.
• Join O to R.

Verification: Measure angle AOR with a protractor. It should be 45 degrees.

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Exercise 14.6 — Angle Bisector and More Constructions

Question 1

Q: Construct angle AOB = 75 degrees using a compass and then bisect it.

Step 1: Draw a ray OA. Step 2: Construct angle AOP = 60 degrees. Step 3: Construct angle POB = 90 degrees on the same side (angle AOB = 90 degrees). Step 4: Bisect angle POB (30 degrees) to get angle POQ = 15 degrees. Step 5: Angle AOQ = 60 + 15 = 75 degrees.

Bisecting 75 degrees: Step 6: With O as centre, draw an arc cutting OA and OQ at points C and D respectively. Step 7: With C as centre and radius greater than half of CD, draw an arc. Step 8: With D as centre and the same radius, draw an arc intersecting the previous arc at E. Step 9: Join O to E.

Angle AOE = 75/2 = 37.5 degrees.

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Question 2

Q: Draw an angle of 140 degrees and bisect it using a compass.

Since 140 degrees cannot be easily constructed with compass alone (it is not a simple combination of 60, 30, 45, 90), we may use a protractor for the initial angle.

Step 1 (using protractor): Draw a ray OA. Place the protractor at O, mark the 140-degree point, and draw ray OB. Angle AOB = 140 degrees.

Bisecting 140 degrees: Step 2: With O as centre, draw an arc cutting OA at P and OB at Q. Step 3: With P as centre and radius greater than half PQ, draw an arc. Step 4: With Q as centre and the same radius, draw an arc intersecting the previous arc at R. Step 5: Join O to R.

Angle AOR = 140/2 = 70 degrees.

Verification: Measure angle AOR with a protractor. It should be 70 degrees.

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Question 3

Q: Copy an angle of 40 degrees using a compass.

Step 1: Let the given angle be POQ = 40 degrees. Step 2: Draw a ray AB on the right side of the page. Step 3: With O as centre, draw an arc cutting OP at C and OQ at D. Step 4: With A as centre and the same radius (OC), draw an arc cutting AB at E. Step 5: With E as centre and radius equal to CD (use compass to measure CD), draw an arc cutting the previous arc at F. Step 6: Join A to F with a ray AF.

Angle BAF = angle POQ = 40 degrees.

Key principle: The angle is copied by transferring the arc distance CD to the new position. This ensures the new angle subtends the same arc as the original.

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Question 4

Q: Construct a rectangle of length 5 cm and breadth 3 cm using a ruler and compass.

ABCD is the required rectangle.

Verification:

• AB = CD = 5 cm (opposite sides equal)
• AD = BC = 3 cm (opposite sides equal)
• All angles = 90 degrees
• Diagonals AC and BD are equal

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Question 5

Q: Construct a square of side 4.5 cm using a ruler and compass.

ABCD is the required square.

Verification:

• AB = BC = CD = DA = 4.5 cm (all sides equal)
• All angles = 90 degrees
• Diagonals AC = BD and they bisect each other at right angles

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Common Mistakes

1. Changing compass opening during construction. When copying a length or an angle, the compass opening must remain fixed. Even a slight change in the compass gap leads to incorrect constructions. Always double-check the compass setting before drawing the final arc.

2. Using the wrong radius for perpendicular bisector. When constructing the perpendicular bisector, the compass radius must be greater than half the line segment. If the radius is too small, the arcs from the two endpoints will not intersect, and the perpendicular bisector cannot be drawn.

3. Confusing the steps for constructing perpendiculars through a point on the line vs. outside the line. When the point is ON the line, we draw arcs from the point to cut the line at two places, then bisect. When the point is OUTSIDE the line, we draw arcs from the point to cut the line at two places, then bisect the segment between those two points. The initial step is similar but the reasoning differs.

4. Constructing angles with wrong arc intersections. When bisecting an angle, the two arcs drawn from the intersection points on the arms must intersect at a single point. If the compass radius is too small, the arcs will not intersect. Always use a radius greater than half the distance between the two points on the arms.

5. Forgetting to verify constructions. Always verify your construction by measuring the final result. For perpendicular bisectors, check that the two segments are equal. For angle bisections, measure the resulting angles. For copied line segments, compare lengths using a compass.

6. Drawing a 90-degree angle incorrectly. A common mistake is to try to construct 90 degrees by bisecting 180 degrees (a straight line), which does not work by simply bisecting with a protractor. The correct compass method involves constructing arcs from equidistant points on the line and joining their intersections.

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Exam Checklist

• [ ] I can identify and properly use the four basic construction tools: ruler, compass, protractor, and divider.
• [ ] I can construct a line segment of a given length using a ruler and compass.
• [ ] I can copy a line segment using a compass without measuring the length numerically.
• [ ] I can construct a circle of given radius and identify its centre, radius, diameter, and chords.
• [ ] I can construct the perpendicular bisector of a line segment and verify that it divides the segment into two equal parts at 90 degrees.
• [ ] I can construct a perpendicular to a line through a point on the line and through a point outside the line.
• [ ] I can construct angles of 60, 30, 90, 120, 150, and 45 degrees using a compass.
• [ ] I can bisect a given angle using a compass and verify the result.
• [ ] I can copy a given angle using a compass (transfer an angle to a new position without measuring it).
• [ ] I can construct a rectangle and a square of given dimensions using a ruler and compass.

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Quick Reference Table: Angle Constructions

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Summary of Construction Steps