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NCERT Solutions Ganita Prakash Chapter 9 & 231Section 9.2 Rotational Symmetry — In-text Questions

Book page 230 Updated on2026-09-05

Q1.
Will the windmill above look exactly the same when rotated through an angle of less than 90°?
Answer

No. The paper windmill has four identical blades placed at equal distances round the centre.

Angle between two neighbouring blades = 360° ÷ 4 = 90°

If you turn it by anything less than 90° — say 30° or 45° or 80° — every blade stops between two old blade positions, so the picture looks tilted and does not match the original.

Only when the turn reaches 90° does each blade land exactly where the next blade was, and the windmill looks untouched.

Why 90° and not less: for the figure to look the same, a blade must land on a blade. The first chance to do that comes after a turn of one whole “gap”, and the gap is 90°. That is why 90° is called the smallest angle of symmetry of the windmill.
Q2.
Thus, we see that the windmill has 4 angles of symmetry. Do you know of any other shape that has exactly four angles of symmetry?
Answer

Yes, many. Any figure built by repeating one piece four times around a centre has exactly 4 angles of symmetry — 90°, 180°, 270°, 360°.

  • A square (and the square tile of a chessboard).
  • A plus sign “+” made of four equal arms, like the figure on page 232.
  • A four-petal flower and the four-pointed star of page 227.
  • A table fan with 4 blades, a ceiling fan with 4 blades, a car steering wheel with 4 equal spokes.
  • Many rangoli designs, and the shape of a playing-card “club” arrangement of 4 identical marks.
Careful: the square has 4 angles of symmetry and 4 lines of symmetry, but the windmill has 4 angles of symmetry and no line of symmetry. Angles and lines are counted separately.
Q3.
How many angles of symmetry does a square have? How much rotation does it require to get the initial square? Do you know where to mark the centre of rotation? What are the other angles of symmetry?
Answer

A square has 4 angles of symmetry.

A B C D → 90° D A B C
A quarter turn (90°) clockwise about the red centre: A goes to B's place, B to C's, C to D's and D to A's. The square looks unchanged.
  • How much rotation is needed? A turn of 90° is enough — the square already covers itself. Each corner simply moves to the next corner's place: A → B, B → C, C → D, D → A.
  • Where is the centre of rotation? At the centre of the square — the point where its two diagonals cross. It is the only point that does not move.
  • The other angles of symmetry:
    90° (quarter turn),   180° (half turn),   270° (three-quarter turn),   360° (full turn)
Why exactly these four: after 180° the corners read C, D, A, B; after 270° they read D, A, B, C; after 360° everything is back where it started. Any other angle would leave the corners between two positions, and the square would look tilted.
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