Symmetry | FIO

Question 15

Draw, wherever possible, a rough sketch of:

a. A triangle with at least two lines of symmetry and at least two angles of symmetry.

b. A triangle with only one line of symmetry but not having rotational symmetry.

c. A quadrilateral with rotational symmetry but no reflection symmetry.

d. A quadrilateral with reflection symmetry but not having rotational symmetry.

Solution

Understand the Question

Line Symmetry (Reflection Symmetry): A figure has line symmetry if it can be folded along a line such that the two halves coincide exactly.
Rotational Symmetry: A figure has rotational symmetry if it looks identical to its original position after a rotation of less than $360^\circ$ around its center.
Order/Angles of Symmetry: The number of times a figure coincides with itself during a complete $360^\circ$ turn determines its order and angles of rotation.

a A triangle with at least two lines of symmetry and at least two angles of symmetry.

Step 1 · Identify the Triangle with Multiple Symmetries

An equilateral triangle satisfies these conditions:

Lines of symmetry: It has $3$ lines of symmetry (along the angle bisectors / medians).
Rotational symmetry: It maps onto itself at rotation angles of $120^\circ, 240^\circ,$ and $360^\circ$ , giving it $3$ angles of symmetry (order of rotational symmetry $= 3$ ).

Answer

a \text{Equilateral Triangle}

b A triangle with only one line of symmetry but not having rotational symmetry.

Step 1 · Identify the Triangle with Single Line Symmetry

An isosceles triangle satisfies these conditions:

Line of symmetry: Exactly $1$ line of symmetry passing from the vertex between the equal sides perpendicular to the base.
Rotational symmetry: It does not match itself at any rotation less than $360^\circ$ (order of rotational symmetry $= 1$ , i.e., no rotational symmetry).

Answer

b \text{Isosceles Triangle}

c A quadrilateral with rotational symmetry but no reflection symmetry.

Step 1 · Identify the Quadrilateral with Rotational Symmetry Only

A parallelogram (that is not a rectangle or a rhombus) satisfies these conditions:

Rotational symmetry: It has rotational symmetry of order $2$ at angles of $180^\circ$ and $360^\circ$ .
Line of symmetry: It has $0$ lines of symmetry (folding along any line does not make the halves coincide).

Answer

c \text{Parallelogram (not a rectangle or rhombus)}

d A quadrilateral with reflection symmetry but not having rotational symmetry.

Step 1 · Identify the Quadrilateral with Line Symmetry Only

An isosceles trapezium (or a kite) satisfies these conditions:

Isosceles Trapezium: Has $1$ line of symmetry connecting the midpoints of the parallel sides, but no rotational symmetry of order $> 1$ .
Kite: Has $1$ line of symmetry along its main diagonal, but no rotational symmetry of order $> 1$ .

Diagram 5

Answer

d \text{Isosceles Trapezium (or Kite)}

Common Mistakes

Assuming all symmetric figures have rotational symmetry: An isosceles triangle or kite has a line of symmetry but no rotational symmetry of order $> 1$ .
Assuming diagonals are lines of symmetry: In a general parallelogram, the diagonals bisect the area, but they are not lines of reflection symmetry.
Order of $1$ : Every shape maps onto itself after a $360^\circ$ rotation. Rotational symmetry is only considered to exist if the order is greater than $1$ (rotation angle $< 360^\circ$ ).

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