NCERT Solutions for Class 5th Maths Chapter 10 The blue design of triangles and circles — Making Designs
Book page 139 Updated on2026-09-19
Q1.
(a) Does the design have rotational symmetry? Yes/No.The design given on page 139, made of circles and triangles.
Answer
No. The design does not have rotational symmetry.
Look at the design carefully first. It is made of a circle in the middle, four triangles round it, and one more small circle on the right.
Notice: the left triangle points inwards, the right one points outwards, and there is only one small circle — on the right.
Test it, step by step:
Try a 1/4 turn. The top triangle would move to the right side. But the right side already has a triangle pointing the other way, and a small circle beyond it. It does not match.
Try a 1/2 turn. The small circle on the right would move to the left. There is nothing on the left for it to land on. It does not match.
Try a 3/4 turn. Again nothing matches.
Only a full turn brings the design back — and a full turn does not count.
1/4 turn → does not match 1/2 turn → does not match 3/4 turn → does not match Full turn → matches, but this is true for every shape So: no rotational symmetry
Why: For rotational symmetry, every piece must have a partner waiting for it in the new place. Here the lone small circle on the right has no partner anywhere. One odd piece is enough to spoil the symmetry.
But it is not symmetry-free: Fold the design along a flat line through the middle. The top triangle lands on the bottom triangle, and the side pieces fold onto themselves. So this design does have reflection symmetry — one horizontal line of symmetry.
Q2.
(b) Try to change the design by adding some shape(s) so that the new design looks the same after a 1/2 turn. Draw the new design in your notebook.The design given on page 139, made of circles and triangles.
Answer
Add one small circle on the left, and turn the right triangle round so that it points inwards like the left one. Then every piece has a partner exactly opposite it.
The red dot is the centre. Turn the picture upside down — it looks exactly the same.
How to build it, step by step:
Copy the design into your notebook.
Mark the centre — the middle of the big circle.
Look at each piece and ask: what is directly opposite it, on the other side of the centre?
Top triangle ↔ bottom triangle. Already a pair. ✓
Left triangle points inwards, but the right one points outwards. Turn the right triangle round so it also points inwards. Now they are a pair. ✓
The small circle on the right has nothing opposite it. Add a small circle on the left, the same size and the same distance away. ✓
Now check: turn the notebook upside down. The design looks unchanged.
Pairs after the change: top triangle ↔ bottom triangle left triangle ↔ right triangle left small circle ↔ right small circle big circle ↔ itself (it sits on the centre)
The rule for a 1/2-turn design: Every piece must have a twin standing straight across the centre, at the same distance, turned the other way. If even one piece is alone, the half turn will show it.
Another correct answer: Instead of turning the right triangle, you may add a triangle at each side so that both sides become bow-tie shapes, and add the left circle. Any design where each piece has an opposite twin is right.
Q3.
(c) Now try to modify or add more shapes so that the new design looks the same after 1/4 turn. Draw the new design in your notebook.The design given on page 139, made of circles and triangles.
Answer
Make all four arms exactly alike: a triangle pointing outwards at the top, right, bottom and left, and a small circle at the end of each arm.
Four identical arms, spaced equally round the centre.
How to build it, step by step:
Draw the big circle in the middle of your page and mark its centre.
Draw one arm — a triangle pointing outwards with a small circle at its tip.
Copy that arm exactly to the right of the centre.
Copy it again below the centre, and once more to the left.
All four arms must be the same size and the same distance from the centre.
Turn your notebook a quarter of the way round. Each arm sits where the next one was, so the design looks unchanged.
Number of equal arms = 4 One full turn ÷ 4 = 1/4 turn So it looks the same at 1/4, 1/2, 3/4 and full turn = 4 times in a full turn
Why 4 equal arms give a 1/4 turn: A quarter turn moves arm 1 to where arm 2 was, arm 2 to arm 3, arm 3 to arm 4 and arm 4 to arm 1. Since all four look alike, you cannot see any change.
Notice: A design that looks the same after a 1/4 turn also looks the same after a 1/2 turn (two quarter turns together). So this design answers part (b) as well.
Q4.
(d) Do the new designs have reflection symmetry? If yes, draw the lines of symmetry.
Answer
Yes — both new designs have reflection symmetry. The 1/2-turn design has 2 lines of symmetry. The 1/4-turn design has 4.
Red dashed lines are the folding lines. Fold along any of them and the two halves match.
Design (b) — check each line:
Upright line through the centre. The left triangle folds onto the right triangle, and the left circle onto the right circle. ✓
Flat line through the centre. The top triangle folds onto the bottom triangle. ✓
So design (b) has 2 lines of symmetry.
Design (c) — check each line:
Upright line: left arm folds onto right arm. ✓
Flat line: top arm folds onto bottom arm. ✓
Slanting line one way: top arm folds onto right arm, left arm onto bottom arm. ✓
Slanting line the other way: top arm folds onto left arm, right arm onto bottom arm. ✓
So design (c) has 4 lines of symmetry.
Why design (c) has more lines: All four arms are exactly alike, so a fold in any of the four directions still lands an arm on an arm. In design (b) the up-down pair and the left-right pair are different from each other, so only two folds work.
Important: Rotational symmetry and reflection symmetry do not always come together. A design can look the same after a 1/4 turn and still have no folding line at all — you will see one such design in the next activity.