**Exercise 18.1 **

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**Question :1. State the number of lines of symmetry for the following figures:**

**(i) An equilateral triangle**

**(ii) An isosceles triangle**

**(iii) A scalene triangle**

**(iv) A rectangle**

**(v) A rhombus**

**(vi) A square**

**(vii) A parallelogram**

**(viii) A quadrilateral**

**(ix) A regular pentagon**

**(x) A regular hexagon**

**(xi) A circle**

**(xii) A semi-circle**** **

**Solution 1:**

(i) An equilateral triangle:

The line of symmetry splits a figure into two parts in such a way that when the figure is folded along the line, the two parts of the figure overlap.

As a result, an equilateral triangle has three symmetry lines.

(ii) An isosceles triangle:

The line of symmetry splits a figure into two parts in such a way that when the figure is folded along the line, the two parts of the figure overlap.

As a result, an isosceles triangle has only one symmetry axis.

(iii) A scalene triangle:

The line of symmetry splits a figure into two parts in such a way that when the figure is folded along the line, the two parts of the figure overlap.

As a result, a scalene triangle has no symmetry rows.

(iv) A rectangle:

As a result, a rectangle has two symmetry lines.

(v) A rhombus:

As a result, a rhombus has two symmetry lines.

(vi) A square:

As a result, a square has four symmetry lines.

(vii) A parallelogram:

As a result, a parallelogram has no symmetry rows.

(viii) A quadrilateral:

As a result, a quadrilateral has no symmetry rows.

(ix) A regular pentagon:

A regular pentagon The line of symmetry splits a figure into two parts in such a way that when the figure is folded along the line, the two parts of the figure overlap.

As a result, a regular pentagon has five symmetry lines.

(x) A regular hexagon:

As a result, a regular hexagon has six symmetry lines.

(xi) A circle:

As a result, a circle has an infinite number of symmetry lines running around its diameters.

(xii) A semi-circle:

As a result, a semicircle has only one symmetry axis.

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**Question :2. What other name can you give to the line of symmetry of**

**(i) An isosceles triangle?**

**(ii) A circle?**** **

**Solution 2:**

(i) There is just one symmetry line in an isosceles triangle. The altitude of an isosceles triangle is also known as this symmetry axis.

(ii) A circle has infinite symmetry lines running the length of its diameters.

**Question :3. Identify three examples of shapes with no line of symmetry.**** **

**Solution 3:**

There is no symmetry line in a scalene triangle, a parallelogram, or a trapezium.

**Question :4. Identify multiple lines of symmetry, if any, in each of the following figures:**

**Solution 4:**

(a) There are three symmetry lines in the specified figure. As a result, it has several symmetry lines.

(b) There are two symmetry lines in the specified diagram. As a result, it has several symmetry lines.

(c) There are three symmetry lines in the specified figure. As a result, it has several symmetry lines.

(d) There are two symmetry lines in the specified diagram. As a result, it has several symmetry lines.

(e) There are four symmetry lines in the specified figure. As a result, it has several symmetry lines.

(f) There is only one symmetry line in the given figure.

(g) There are four symmetry lines in the specified figure. As a result, it has several symmetry lines.

(h) There are six symmetry lines in the specified figure. As a result, it has several symmetry lines.

**Exercise 18.2**

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**Question :1. In the following figures, the mirror line (i.e. the line of symmetry) is given as dotted line. Complete each figure performing reflection in the dotted (mirror) line. Also, try to recall name of the complete figure.**

**Solution 1:**** **

(a) It will be in the form of a rectangle.

(b) It will be in the form of a triangle.

(c) It will be in the form of a rhombus.

(d) It will be in the form of a circle.

(e) It will be in the form of a pentagon.

(f) It will be in the form of an octagon.

**Question :2. Each of the following figures shows paper cuttings with punched holes. Copy these figures on a plane sheet and mark the axis of symmetry so that if the paper is folded along it, then the wholes on one side of it coincide with the holes on the other side.**

**Solution 2:**

The lines of symmetry in the given figures are as follows:

**Question :3. In the following figures if the dotted lines represent the lines of symmetry, find the other hole(s).**

**Solution 3:**

The following are the other holes in the figure:

**Exercise 18.3**** **

**Question :1. Give the order of rotational symmetry for each of the following figures when rotated about the marked point (x):**

**Solution 1:**

(i) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 4.

(ii) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 3.

(iii) If a figure fits into itself more than once during a complete rotation of 360^{o}, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is equal to three.

(iv) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 4.

(v) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 2.

(vi) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 4.

(vii) If a figure fits into itself more than once during a complete rotation of 360^{o}, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is equal to four.

(viii) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 6.

(ix) When a figure fits into itself more than once during a complete 360^{o} rotation, it is said to have rotational symmetry.

As a result, the rotational symmetry of the given figure is 3.

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**Question :2. Name any two figures that have both line symmetry and rotational symmetry.**** **

**Solution 2:**

Both lines of symmetry and rotational symmetry exist in an equilateral triangle and a square.

**Question :3. Give an example of a figure that has a line of symmetry but does not have rotational symmetry.**** **

**Solution 3:**

A line of symmetry exists in a semicircle and an isosceles triangle, but rotational symmetry does not exist.

**Question :4. Give an example of a geometrical figure which has neither a line of symmetry nor a rotational symmetry.**** **

**Solution 4:**

There is no line of symmetry or rotational symmetry in a scalene triangle.

**Question :5. Give an example of a letter of the English alphabet which has**

**(i) No line of symmetry**

**(ii) Rotational symmetry of order 2.**** **

**Solution 5:**

(i) No line of symmetry:

Z is the only letter in the English alphabet that does not have a symmetry line.

(ii) Rotational symmetry of order 2:

The letter N is the only letter in the English alphabet with rotational symmetry of order two.

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**Question :6. What is the line of symmetry of a semi-circle? Does it have rotational symmetry?**** **

**Solution 6:**

There is just one symmetry line in a semicircle (half of a circle). There is just one symmetry line in the illustration. Along the perpendicular bisector I of the diameter XY, the figure is symmetric. There is no rotational symmetry in a semi-circle.

**Question :7. Draw, whenever possible, a rough sketch of**

**(i) a triangle with both line and rotational symmetries.**

**(ii) a triangle with only line symmetry and no rotational symmetry.**

**(iii) a quadrilateral with a rotational symmetry but not a line of symmetry.**

**(iv) a quadrilateral with line symmetry but not a rotational symmetry.**** **

**Solution 7:**

(i) An equilateral triangle has three lines of symmetry and order three rotational symmetry.

(ii) There is just one line of symmetry in an isosceles triangle, and there is no rotational symmetry.

(iii) A parallelogram is a quadrilateral with rotational symmetry of order 2 but no line of symmetry.

(iv) A quadrilateral with just one line of symmetry and no rotational symmetry is called a kite.

**Question :8. Fill in the blanks:**

**Solution 8:**

**Question :9. Fill in the blanks:**

**Solution 9:**