Symmetry | IT

Question 14

Draw the following.

a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry.

Is it possible to draw a triangle with exactly two lines of symmetry?

Solution

Understand the Question

A line of symmetry is a line that divides a figure into two identical mirror-image halves that match perfectly when folded.
Based on side lengths, triangles are categorized as:
- Equilateral triangle: All $3$ sides equal $\rightarrow 3$ lines of symmetry
- Isosceles triangle: $2$ sides equal $\rightarrow 1$ line of symmetry
- Scalene triangle: All sides unequal $\rightarrow 0$ lines of symmetry
A triangle can only have $0$ , $1$ , or $3$ lines of symmetry—never exactly $2$ .

a. A triangle with exactly one line of symmetry.

Step 1 · Draw an Isosceles Triangle

An isosceles triangle has two sides of equal length and two equal angles. Diagram 1

The line drawn from the vertex between the equal sides perpendicular to the opposite base divides the triangle into two matching halves. Hence, it has exactly $1$ line of symmetry.

Answer

a. An isosceles triangle has exactly one line of symmetry.

b. A triangle with exactly three lines of symmetry.

Step 1 · Draw an Equilateral Triangle

An equilateral triangle has all three sides of equal length and all three angles equal ( $60^\circ$ ). Diagram 2

A line drawn from each vertex to the midpoint of the opposite side is a line of symmetry. Hence, it has exactly $3$ lines of symmetry.

Answer

b. An equilateral triangle has exactly three lines of symmetry.

c. A triangle with no line of symmetry.

Step 1 · Draw a Scalene Triangle

A scalene triangle has all three sides of different lengths and all three angles of different measures.No line can divide the triangle into two identical halves when folded. Hence, it has no lines of symmetry ( $0$ lines of symmetry).

Answer

c. A scalene triangle has no line of symmetry.

Is it possible to draw a triangle with exactly two lines of symmetry?

Step 1 · Analyze Possibility of Two Lines of Symmetry

If a triangle has at least one line of symmetry, it must have at least two equal sides (isosceles).
If it has a second line of symmetry, another pair of sides must also be equal, making all three sides equal (equilateral).
An equilateral triangle automatically possesses $3$ lines of symmetry, not $2$ .

Therefore, it is impossible for a triangle to have exactly $2$ lines of symmetry.

Answer

No, it is not possible to draw a triangle with exactly two lines of symmetry.

Common Mistakes

Assuming 2 Lines of Symmetry Exist: Rectangles and rhombuses have $2$ lines of symmetry, but triangles cannot. As soon as a triangle gains a second line of symmetry, all three sides become equal, giving it $3$ lines of symmetry.
Confusing Isosceles and Scalene: An isosceles triangle has $1$ line of symmetry, while a scalene triangle has $0$ lines of symmetry.

More questions in IT

What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?

Can you see what repeats in the beautiful rangoli figure?

What about the pinwheel? Can you spot which pattern is repeating?

Hint: Look at the hexagon first.

Now, can you say what figure repeats along each side of the hexagon? What is the shape of the figure that is stuck to each side? Do you recognise it? How do these shapes move as you move along the boundary of the hexagon? What about the other pictures—what is it about those structures that appeals to you and what are the patterns in those structures that repeat?

What are the symmetries that you see in these beautiful structures?

Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?

Thus, figures can have multiple lines of symmetry. The figures below also have multiple lines of symmetry. Can you find them all?

We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry?

First, see the rectangle and answer this question. Then, take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. What do you observe?

Context: Consider a square with its corners labeled $A$ , $B$ , $C$ and $D$ , and its vertical line of symmetry as shown in the figure.

Q. What if we reflect along the diagonal from $A$ to $C$ ? Where do points $A$ , $B$ , $C$ and $D$ go? What if we reflect along the horizontal line of symmetry?

Q10

In these two figures, a sheet of paper is folded and a cut is made along the dotted line shown. Draw a sketch of how the paper will look when unfolded.

Do you see a line of symmetry in this figure? What is it?

Q11

5 Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?

a. The hole in the centre is a square.

b. The hole in the centre is a square.

Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.

Q12

How many lines of symmetry do these shapes have?

Q13

Find the lines of symmetry for the kolam below.

Q14

Draw the following.

a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry.

Is it possible to draw a triangle with exactly two lines of symmetry?

Q15

Draw the following. In each case, the figure should contain at least one curved boundary.

a. A figure with exactly one line of symmetry. b. A figure with exactly two lines of symmetry. c. A figure with exactly four lines of symmetry.

Q16

Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you.

Hint: For (c) and (f), see if rotating the book helps!

Q17

Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.

Q18

Do you know of any other shape that has exactly four angles of symmetry?

Q19

Consider this figure, a picture with 4 radial arms. How many angles of symmetry does it have? What are they? Note that the angle between adjacent central dotted lines is $90^\circ$ .

Q20

Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.

Q21

How will you modify the figure above so that it has only two angles of symmetry?

Q22

Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this?

Q23

Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they?

Q24

However, can anything in the figure be changed to make it have 3 angles of symmetry?

Q25

Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.

Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?

Q26

Consider a figure with radial arms having exactly $7$ angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.

Q27

In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?

Q28

True or False

Every figure will have $360^\circ$ as an angle of symmetry.
If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of $360$ .

Q29

Like wheels, we can find other objects around us having rotational symmetry. Find them.