NCERT Solutions Ganita Prakash (Part 1) Chapter 5 –114Section 5.4 Parallel and Perpendicular Lines in Paper Folding — Figure it Out
Book page 113 Updated on2026-09-05
Q1.
Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.
Answer
Read each given segment as a step: "so many dots right, so many dots down". To turn it, swap the two numbers and change one sign.
Given segment
Its step
A perpendicular segment
Horizontal one, top row
2 right, 0 down
0 right, 1 down — a vertical segment
Slanting one, bottom left
2 right, 2 up
2 right, 2 down — the other 45° slant
Vertical one, middle
0 right, 2 down
2 right, 0 down — a horizontal segment
Steep one, right side
2 right, 3 down
3 right, 2 up
Grey = the four given lines; blue = the perpendiculars drawn on the dots. Each blue segment crosses its grey partner at 90°, and every endpoint sits on a dot.
Why it happens: If one segment goes p right and q down, a segment that goes q right and pup turns the direction by exactly a quarter turn. The dot grid keeps both endpoints on dots, so you never need a protractor.
Check it yourself: Put the corner of a page at each crossing. It should fit snugly with no gap.
Q2.
In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol. (a) How did you spot the perpendicular lines? (b) How did you spot the parallel lines?
Answer
Work shape by shape on the grid.
Mark every vertical edge with a single arrow head (>), and every horizontal edge with a double arrow head (>>).
The slanting edges that run along the grid diagonals fall into two families — down-to-the-right and up-to-the-right. Mark one family with three arrow heads and the other with four.
Wherever a vertical edge meets a horizontal edge, draw the small square symbol for 90°.
(a) How did you spot the perpendicular lines? The vertical and horizontal lines of the grid paper meet at a right angle (90°), so wherever one edge follows a grid column and the other follows a grid row, the two edges are perpendicular. The corner of a page or a set square confirms it.
(b) How did you spot the parallel lines? By finding lines that always stay the same distance apart. On grid paper this is easy: two edges are parallel when they take the same step — for example both go 1 square right and 1 square down.
Why it happens: A grid is itself made of two families of parallel lines that cut each other at 90°. Any edge drawn along the grid inherits these relationships, so you can judge parallel and perpendicular by counting squares instead of measuring.
Q3.
In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.
Answer
Choose a step and repeat it starting from different dots. Some easy sets are given below.
Set
Step used
Example segments (dots joined)
1
3 right, 0 down
Three horizontal segments of lengths 2, 3 and 4 units in different rows
2
0 right, 2 down
Vertical segments in different columns
3
1 right, 1 down
45° segments starting from different dots
4
2 right, 1 up
Gently rising segments — a harder but neat set
Why it happens: Two segments are parallel exactly when they have the same direction. On dot paper the direction is completely fixed by the step, so same step = parallel, whatever the length and wherever you start.
Tip: Make the segments different lengths on purpose. It reminds you that being parallel is about direction, not size.
Q4.
Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper. (a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it?
Answer
For each segment in Fig. 5.12, count its step and repeat that same step from a different dot.
(a) Did you find it challenging?Yes — some were much harder than others.
(b) Which ones? The segments e, f, g and h. These do not run along a row, a column or a 45° diagonal, so the eye cannot judge them easily. The horizontal, vertical and 45° ones (a, b, c, d) were simple.
(c) How did you do it? Two reliable methods:
Count the step. If the segment goes, say, 3 dots right and 1 dot down, start at any other dot and go 3 right and 1 down. Join the two dots.
Keep the distance equal. Mark two points at the same perpendicular distance from the given segment, one near each end, and join them.
Why it happens: A slanted direction like "3 right, 1 down" has no landmark on the page to copy, so guessing goes wrong. Counting dots turns the direction into two whole numbers, which are easy to repeat exactly.
Check it yourself: Draw any line cutting both segments. If your drawing is right, the two angles it makes in the same position at the two crossings will be equal.
Q5.
In Fig. 5.13, which line is parallel to line a — line b or line c? How do you decide this?
Answer
Line c is parallel to line a.
Lines b and c start from the same point, and in the figure there is already a segment joining that point to the end of line a. Use it as a transversal and compare the angles it makes.
Angle between the joining line and line a ≈ 29° Angle between the joining line and line c ≈ 29° → equal ✔ Angle between the joining line and line b ≈ 32° → not equal ✘
Corresponding angles are equal only for c and a, so c ∥ a. Line b leans a little more, so b and a would meet if extended far enough to the right.
Why it happens: Eyes are poor at judging small differences in slope — 29° and 32° look alike. A transversal converts the question into a comparison of two angles, which a protractor (or tracing paper) settles exactly.
Tip: Another quick check — measure the perpendicular gap between a and c near both ends. For a parallel pair the two readings match.