Question 17
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.
- A line of symmetry acts like a mirror line, dividing a figure into two identical halves that coincide when folded along the line.
- Since the completed drawing must have two perpendicular lines of symmetry (the horizontal and vertical blue lines):
- Every point must be reflected across the vertical blue line (left-right reflection).
- Every point must also be reflected across the horizontal blue line (top-bottom reflection).
- This ensures that each quadrant mirrors the adjacent quadrants precisely across the blue lines.
(a) Complete drawing (a) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the horizontal red line across the horizontal blue line of symmetry.
- The given horizontal segment lies below the horizontal blue line, extending to the left and to the right of the vertical line.
- Since it is already symmetric about the vertical line, reflecting it across the horizontal blue line creates an identical segment above the horizontal line.
(a) Completed figure as shown in the diagram.
(b) Complete drawing (b) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the zig-zag shape across both the vertical and horizontal blue lines.
- Reflect the given red line across the vertical blue line to draw its mirror image on the right half.
- Reflect the entire drawing across the horizontal blue line to obtain mirror images in the lower quadrants, completing the full symmetric figure.
(b) Completed figure as shown in the diagram.
(c) Complete drawing (c) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the zig-zag line in the top-left quadrant across both blue lines.
- Reflect the top-left zig-zag line across the vertical blue line to create its mirror image in the top-right quadrant.
- Reflect the entire upper half across the horizontal blue line to create mirror images in the bottom-left and bottom-right quadrants.
(c) Completed figure as shown in the diagram.
(d) Complete drawing (d) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the polygonal shape in the bottom-left quadrant across both lines of symmetry.
- Reflect the bottom-left polygon across the vertical blue line to produce its mirror image in the bottom-right quadrant.
- Reflect both lower sections across the horizontal blue line into the top-left and top-right quadrants to close the symmetric polygon.
(d) Completed figure as shown in the diagram.
(e) Complete drawing (e) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the shape in the top-right quadrant across both lines of symmetry.
- Reflect the top-right polygon across the vertical blue line into the top-left quadrant.
- Reflect the upper half across the horizontal blue line into the bottom-right and bottom-left quadrants.
(e) Completed figure as shown in the diagram.
(f) Complete drawing (f) so that the resulting figure has the two blue lines as lines of symmetry.
Step 1 · Reflect across Lines of Symmetry
Reflect the polygon in the bottom-right quadrant across both lines of symmetry.
- Reflect the bottom-right polygon across the vertical blue line into the bottom-left quadrant.
- Reflect both bottom sections across the horizontal blue line to draw the corresponding mirror images in the top-right and top-left quadrants.
(f) Completed figure as shown in the diagram.
- Single-axis reflection: Reflecting across only one blue line (e.g., only horizontal or only vertical) instead of both lines of symmetry.
- Grid counting error: Not measuring the exact distance (number of squares) of each vertex from the blue line, causing the reflected shape to be distorted.
- Slope/direction inversion error: Drawing parallel lines instead of flipping the direction/slope of segments across the mirror line.
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 , , and , and its vertical line of symmetry as shown in the figure.
Q. What if we reflect along the diagonal from to ? Where do points , , and go? What if we reflect along the horizontal line of symmetry?
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?
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.
- How many lines of symmetry do these shapes have?
Find the lines of symmetry for the kolam below.
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?
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.
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!
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.
Do you know of any other shape that has exactly four angles of symmetry?
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 .
Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
How will you modify the figure above so that it has only two angles of symmetry?
Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this?
Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they?
However, can anything in the figure be changed to make it have 3 angles of symmetry?
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?
Consider a figure with radial arms having exactly 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.
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?
True or False
- Every figure will have 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 .
Like wheels, we can find other objects around us having rotational symmetry. Find them.