GSEB Class 6 Maths Solutions Chapter 14 Practical Geometry Exercise 14.3

Question 1. Draw any line segment \( \overline{P Q} \). Without measuring \( \overline{P Q} \), construct a copy of \( \overline{PQ} \).
Answer:
Steps of construction:
Step I: First, draw a line segment \( \overline{XY} \) whose exact length is not yet known.
Step II: Next, draw a line `l` and put a point `R` on it.
Step III: Open your compasses to match the exact length of \( \overline{XY} \).
Step IV: Keeping the compasses open to the same size, place the metal point of the compasses at `R` and make a mark for point `S` on the line `l`.
So, \( \overline{RS} \) is an exact copy of \( \overline{XY} \).

Exam Tip: Remember that copying a line segment means creating another segment of the same exact length using only a compass and a straightedge, without any direct measurement with a ruler.

Question 2. Draw any line segment \( \overline{AB} \), whose length you do not know. Construct a line segment \( \overline{PQ} \) such that the length of \( \overline{PQ} \) is twice that of \( \overline{AB} \).
Answer:
Steps of construction:
Step I: First, draw a line segment \( \overline{AB} \) of unknown length.
Step II: Place the steel point of your compasses at point `A` and the pencil end at point `B`. This opening of the compasses now represents the length of \( \overline{AB} \).
Step III: Next, draw a line `l` and mark a point `P` on it.
Step IV: Without altering the compasses' opening, place the steel point on `P` and make a mark for point `R` on line `l`. This means \( PR = AB \).
Step V: Now, place the steel point at `R` and, keeping the same opening, make another mark for point `Q` on line `l`. This means \( RQ = AB \).
Therefore, \( \overline{PQ} \) is the necessary line segment whose total length is double that of \( \overline{AB} \), i.e., \( \overline{PQ} = 2 \overline{AB} \).

Exam Tip: When doubling a line segment, ensure you make two consecutive marks of the original length on the new line without changing the compass opening. The final segment will span from the first point to the second mark.