GSEB Class 6 Maths Solutions Chapter 14 Practical Geometry Exercise 14.2

Question 1. Draw a line segment of length 7.3 cm using a ruler.
Answer: Steps for construction:
Step I: First, mark a point 'A'.
Step II: Next, position the zero mark of your ruler against point 'A'.
Step III: Then, mark another point 'B' at a distance of 7.3 cm from point 'A'.
Step IV: Finally, connect points 'A' and 'B'. Thus, \( \overline{A B} \) is the required line segment with a length of 7.3 cm. Always look straight down at the measuring device when marking points to avoid errors in length.

Exam Tip: When drawing line segments with a ruler, ensure your eye is directly above the mark to prevent parallax errors, which can lead to inaccurate measurements.

Question 2. Construct a line segment of length 5.6 cm using a ruler and compasses.
Answer: Solution: Steps for construction:
Step I: Draw a line 'l' and mark a point 'A' on it.
Step II: Place the steel end of the compasses on the zero mark of the ruler. Open the compasses so the pencil tip reaches the 5.6 cm mark.
Step III: Without altering the compasses' opening, place the steel end on point 'A' and make an arc to cut line 'l' at 'B'. Thus, \( \overline{A B} \) = 5.6 cm is the line segment of the desired length.
In simple words: First, draw a line and pick a starting point. Then, use your ruler to set the compass to the exact length you need. Finally, use the compass to mark that length on your line.

Exam Tip: For constructions involving compasses, make sure the compass opening is precise and does not change accidentally between steps. This is crucial for accurate lengths.

Question 3. Construct \( \overline{AB} \) of length 7.8 cm. From this, cut-off \( \overline{AC} \) of length 4.7 cm. Measure \( \overline{BC} \).
Answer: Solution: Steps for construction:
Step I: Place the ruler's zero mark at point 'A'.
Step II: Mark a point 'B' at a distance of 7.8 cm from 'A'.
Step III: Mark another point 'C' between 'A' and 'B' at a distance of 4.7 cm from 'A' so that \( \overline{A C} \) = 4.7 cm.
Step IV: Measure the line segment \( \overline{BC} \). We discover that \( \overline{BC} \) = 3.1 cm.
In simple words: Draw a long line segment. Then, mark a shorter part from its start. The remaining part is the answer you need to find.

Exam Tip: When cutting off a segment, ensure the second mark is *between* the first two points. The final length should be the difference between the initial segment and the cut-off part.

Question 4. Given \( \overline{AB} \) of length 3.9 cm, construct \( \overline{P Q} \) such that the length of \( \overline{P Q} \) is twice that of \( \overline{AB} \). Verify by measurement.
Answer: Hint: Construct \( \overline{P X} \) such that its length equals the length of \( \overline{A B} \); then cut-off \( \overline{X Q} \) so that \( \overline{X Q} \) also has the length of \( \overline{A B} \).
Solution: Steps for construction:
Step I: Draw a line 'l'.
Step II: Draw \( \overline{AB} \) = 3.9 cm.
Step III: On line 'l', construct \( \overline{P X} = \overline{A B} \) (= 3.9 cm).
Step IV: Next, construct \( \overline{X Q} = \overline{AB} \) (=3.9 cm).
Thus, the lengths of \( \overline{P X} \) and \( \overline{X Q} \) are combined to create a segment twice the length of \( \overline{AB} \), forming \( \overline{PQ} \).
Verification: By measurement, we get:
\( \overline{A B} + \overline{AB} = 3.9 \text{ cm} + 3.9 \text{ cm} \)
\( 2 (\overline{AB}) = 7.8 \text{ cm} = \overline{PQ} \)
Thus, twice of \( \overline{A B} \) is equal to \( \overline{PQ} \).
In simple words: To double a line segment, first draw the original length. Then, immediately after it, draw another segment of the exact same length. The total new line will be twice as long.

Exam Tip: When doubling a segment with a compass, ensure you use the exact same opening for both parts of the new segment. This ensures accuracy in the final length.

Question 5. Given length 7.3 cm and \( \overline{CD} \) of length 3.4 cm, construct a line segment \( \overline{X Y} \) such that the length of \( \overline{X Y} \) is equal to the difference between the lengths of \( \overline{A B} \) and \( \overline{CD} \). Verify by measurement.
Answer: Solution:
Step I: Draw \( \overline{AB} \) = 7.3 cm and \( \overline{CD} \) = 3.4 cm.
Step II: Draw a line 'l' and take a point 'X' on it.
Step III: Construct \( \overline{X R} \) such that its length equals the length of \( \overline{A B} \) (= 7.3 cm).
Step IV: Cut-off \( \overline{RY} \) = length of \( \overline{CD} \) (= 3.4 cm) from \( \overline{XR} \) such that the remaining length \( \overline{X Y} \) = length of \( \overline{A B} \) – length of \( \overline{CD} \).
Verification: By measurement, we find:
\( \overline{X Y} = 3.9 \text{ cm} = 7.3 \text{ cm} – 3.4 \text{ cm} \)
\( = \overline{AB} - \overline{C D} \)
Thus, we get \( \overline{X Y} = \overline{AB} – \overline{CD} \).
In simple words: Draw a line as long as the first measurement. Then, from one end of that line, cut off a part that is the same length as the second measurement. The part left over is your answer.

Exam Tip: When constructing a difference, always start with the longer segment and subtract the shorter one. Precise compass and ruler usage is essential for an accurate result.