ICSE Class 10 Maths Chapter 15 Symmetry

Unit 4 - Geometry

Chapter 15 - Symmetry

Points To Remember

1. Line of Symmetry

Trace the given figure and the dotted line on a piece of paper and fold it along the dotted line. You will find that the two parts of the figure on both sides of the line coincide with each other.

Thus, the dotted line divides the given figure into two identical figures. We say that the given figure is symmetrical about the dotted line.

Line of symmetry. Whenever a line divides a given figure into two congruent figures, i.e., two identical halves, we say that the given figure is symmetrical about that line. And, in this case the line is called the axis of symmetry or line of symmetry.

Some Examples

1. An angle with equal arms is symmetrical about its bisector.

Let \(\angle AOB\) be a given angle with equal arms OA and OB and let OC be the bisector of \(\angle AOB\).

Clearly, OC divides \(\angle AOB\) into two identical halves.

\(\therefore\) OC is the line of symmetry of \(\angle AOB\).

2. An isosceles triangle has one line of symmetry, namely the bisector of the vertical angle.

Let \(\triangle ABC\) be an isosceles triangle with AB = AC.

Let AD be the bisector of \(\angle A\).

Then, \(AD \perp BC\) and D is the mid-point of BC.

Now, in \(\triangle s\) ABD and ACD, we have

\(AB = AC, BD = DC\) and \(AD = AD\).

\(\therefore \triangle ABD \cong \triangle ACD\)

Thus, AD divides \(\triangle ABC\) into two congruent triangles.

\(\therefore\) AD is the line of symmetry of \(\triangle ABC\).

3. An equilateral triangle has three lines of symmetry, namely the bisectors of each of its angles.

Let \(\triangle ABC\) be an equilateral triangle and let AD, BE and CF be the bisectors of \(\angle A, \angle B\) and \(\angle C\) respectively.

Teacher's Note

Symmetry is visible everywhere in nature - from butterfly wings to flower petals. Understanding line symmetry helps students recognize patterns in the world around them.

Clearly, AD divides \(\triangle ABC\) into two congruent triangles.

\(\therefore\) AD is the line of symmetry of \(\triangle ABC\).

Similarly, BE as well as CF is the line of symmetry of \(\triangle ABC\).

4. An isosceles trapezium has one line of symmetry, namely the perpendicular bisector of its parallel sides.

Let ABCD be an isosceles trapezium in which \(AB \parallel DC\) and \(AD = BC\).

Let PQ be a line which is the perpendicular bisector of AB as well as DC.

Clearly, PQ divides ABCD into two identical halves.

\(\therefore\) PQ is the line of symmetry of trap. ABCD.

5. A kite has the vertical diagonal as the line of symmetry.

Let ABCD be the kite in which \(AB = AD\) and \(BC = DC\).

Clearly, the vertical diagonal AC divides the kite ABCD into two identical halves.

\(\therefore\) AC is the line of symmetry of kite ABCD.

6. The vertical diagonal of an arrowhead is the line of symmetry.

Let ABCD be an arrowhead in which \(AB = AD\) and \(BC = DC\).

Clearly, the line AC divides the figure into two identical halves.

Hence, AC is the line of symmetry of the arrowhead ABCD

7. A rectangle has two lines of symmetry, namely each of the lines joining the mid-points of its opposite sides.

Let ABCD be a rectangle and let PQ be the line joining the mid-points of one pair of opposite sides AD and BC.

Clearly, PQ divides rect. ABCD into two identical halves.

\(\therefore\) PQ is a line of symmetry of rect. ABCD.

Similarly, if R and S be the mid-points of DC and AB respectively then RS is a line of symmetry of rect. ABCD.

8. A square has 4 lines of symmetry, namely the two diagonals and the lines joining the mid-points of its opposites sides.

Let ABCD be a square in which \(l_1\) and \(l_2\) be the diagonals ; \(l_3\) be the line joining the mid-points of AB and DC and \(l_4\) be the line joining the mid-points of AD and BC.

Each one of \(l_1, l_2, l_3\) and \(l_4\) divides the sq. ABCD into two identical halves.

\(\therefore\) Each one of these lines is the line of symmetry of sq. ABCD.

Teacher's Note

A square is the most symmetrical quadrilateral, combining the symmetry properties of rectangles and rhombuses. This makes it a perfect example for discussing multiple lines of symmetry.

9. A parallelogram has no line of symmetry.

Let ABCD be a parallelogram.

Clearly, no line divides it into two identical halves.

So, \(\parallel\) gm ABCD has no line of symmetry.

10. A rhombus has two lines of symmetry, namely each of its diagonals.

Let ABCD be a rhombus.

Clearly, the diagonal AC divides the rhombus into two identical halves.

\(\therefore\) AC is a line of symmetry.

Similarly, diagonal BD is a line of symmetry.

11. A circle has an infinite number of lines of symmetry, namely each line passing through the centre of the circle.

Clearly, each diameter of divides the circle into two identical halves.

12. A regular pentagon has 5 lines of symmetry, namely the perpendiculars from vertices to the opposite sides.

Let ABCDE be a regular pentagon.

Let \(l_1\) be a line drawn from A, perpendicular to the opposite side CD.

Clearly, \(l_1\) divides the given pentagon into two identical figures.

\(\therefore l_1\) is the line of symmetry of the given pentagon.

Similarly, each of the lines \(l_2, l_4, l_4\) and \(l_5\) shown in the figure is the line of symmetry of the given pentagon.

13. A regular hexagon has 6 lines of symmetry, namely three diagonals \(l_1, l_2, l_3\) and three lines \(l_4, l_5, l_6\) each joining the mid-points of opposite sides of the hexagon.

In general, a regular polygon of n sides has n lines of symmetry.

Teacher's Note

Regular polygons showcase mathematical beauty through their perfect symmetry. The number of lines of symmetry in a regular polygon equals the number of sides, demonstrating the relationship between shape and symmetry.

14. (i) Each of the letters given below has one line of symmetry, shown by the dotted line.

A - B - C - D - E

M - T - U - V - W - Y

(ii) Each of the letters given below has two lines of symmetry, shown by dotted lines.

H - I - O - X

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