Fractals and Visualising Solids | IT

Question 2

Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.

[Hint: Note that the corner triangles are isosceles.]

This fractal is called the Sierpinski Triangle/Gasket.

Solution

Connecting the middle points of each side of an equilateral triangle creates four smaller triangles inside.

Step 1 — Starting with an equilateral triangle

Let us consider a large equilateral triangle. All its sides are equal in length. All its angles are equal to $60^\circ$ . Let the side length of this large triangle be $2L$ .

Diagram 1

Step 2 — Marking the midpoints

We find the midpoint of each side. A midpoint divides a side into two equal parts. Let D be the midpoint of side AB. Let E be the midpoint of side BC. Let F be the midpoint of side CA.

The length of AD is half of AB. $AD = \frac{1}{2} \times AB$ $AD = \frac{1}{2} \times 2L$

$\boxed{AD = L}$

Similarly, $DB = L$ , $BE = L$ , $EC = L$ , $CF = L$ , and $FA = L$ .

Diagram 2

Step 3 — Forming the inner triangle

We connect these midpoints to form a new triangle. This new triangle is DEF. A line segment joining two midpoints is half the length of the third side. It is also parallel to the third side.

Consider the segment DE. It connects midpoint D of AB and midpoint E of BC. So, DE is half the length of AC. $DE = \frac{1}{2} \times AC$ $DE = \frac{1}{2} \times 2L$

$\boxed{DE = L}$

Similarly, EF connects midpoints E and F. So, EF is half the length of AB. $EF = \frac{1}{2} \times AB = L$

And FD connects midpoints F and D. So, FD is half the length of BC. $FD = \frac{1}{2} \times BC = L$

Diagram 3

Step 4 — Showing all four triangles are equilateral

We now have four triangles: ADF, BDE, CEF, and DEF. Let us look at triangle DEF first. We found that $DE = L$ , $EF = L$ , and $FD = L$ . Since all three sides are equal, triangle DEF is an equilateral triangle.

Now consider triangle ADF. We know $AD = L$ and $AF = L$ . The angle at vertex A is $60^\circ$ . This is because the large triangle ABC is equilateral. Since two sides are equal and the angle between them is $60^\circ$ , the triangle must be equilateral. So, triangle ADF is an equilateral triangle.

Similarly, for triangle BDE: $BD = L$ , $BE = L$ . Angle B is $60^\circ$ . So, triangle BDE is an equilateral triangle.

And for triangle CEF: $CE = L$ , $CF = L$ . Angle C is $60^\circ$ . So, triangle CEF is an equilateral triangle.

All four triangles are equilateral triangles.

Diagram 4

Step 5 — Showing all four triangles are identical

We have shown that all four triangles are equilateral. Each of these triangles has a side length of $L$ . For example, triangle ADF has sides $L, L, L$ . Triangle BDE has sides $L, L, L$ . Triangle CEF has sides $L, L, L$ . Triangle DEF has sides $L, L, L$ . Since they are all equilateral and have the same side length, they are identical. This means they are congruent (exactly the same shape and size).

Therefore, joining the midpoints of an equilateral triangle divides it into 4 identical equilateral triangles.

Answer

(i) Joining the midpoints of an equilateral triangle creates four smaller triangles. (ii) Each of these four smaller triangles is an equilateral triangle. (iii) All four smaller equilateral triangles are identical to each other.

More questions in IT

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.

By its construction, each step in the sequence has (i) squares of the same size that remain in the figure, and the size of these squares becomes smaller and smaller as the step number increases, and (ii) square holes that are formed by removing square pieces.

Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.

[Hint: Note that the corner triangles are isosceles.]

This fractal is called the Sierpinski Triangle/Gasket.

In previous classes, you've seen solids that are much simpler than an elephant or cat, such as cubes, spheres, cylinders, and cones. What would the profiles of these look like, from different viewpoints?

Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can imagine the solid passing through a wall like Tom did, and leaving a hole of the appropriate shape.

A solid whose profile has a square outline
A solid whose profile has a circular outline
A solid whose profile has a triangular outline

As we saw with the elephant, a given solid might have very different profiles from different viewpoints. Can you visualise solids that have the following contrasting profiles?

Spend some time on this, and if you are finding it difficult to visualise, you may look around and use objects that are around you, or that you will make in the next section. Feel free to consider viewpoints from any direction, including directly above the object.

A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint
A solid with a circular profile from one viewpoint and a triangular one from another viewpoint
A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint
A solid with a trapezium shaped profile from one viewpoint and a circular one from another viewpoint
A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint

Are there unique solids for each of the conditions, or can you come up with multiple possibilities?

If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if the polygons have $n$ sides?

If the base of a pyramid has 10 sides, how many faces, edges and vertices does the pyramid have? What if the base is an $n$ -sided polygon?

What is a net of a cube?

Visualise how it can be folded to form a cube.

Q10

What is a net of a regular tetrahedron? Which of the following are nets of a regular tetrahedron?

Q11

Are there any other possible nets?

Q12

Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making an actual cutout.

Q13

Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.

Q14

What is the net of a cylinder?

If the circular faces of a cylinder are unfolded, and if a cut is made along the height of the cylinder, as shown in the figure below, then we get

Q15

What are the sidelengths of the rectangle obtained?

Q16

How will the net of a cone look?

Q17

If the cone is slit open along the line $l$ and then unrolled, what will we get?

Observe that all the points on the boundary of the base circle are at equal distances from 'O'. So after unrolling the cone, the boundary of the net will be a portion of a circle with centre O.

Q18

What surface do you construct by using the above net, in which O is not the centre of the boundary circle? Make a physical model to help you answer this question!

Q19

Draw a net with appropriate measurements that can be folded into a triangular prism. Verify that it works by making an actual cutout.

Q20

Taking all the triangles in the net to be equilateral, make a cutout of the net and fold it to form an octahedron.

Q21

What is the shortest path for the ant to reach the laddu?

Q22

What about in the following case?

Q23

If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities?

Q24

For example, are either of these the shortest path?

Q25

What does this show?

Q26

Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?

Q27

What is the length of the shortest path between the ant and the laddu?

Q29

What happens to the length of a line in its projection?

Q30

Can you now compare the lengths $p$ and $l$ ?

Q31

When is the length of the projected line equal to its actual length?

Q32

What do you think are the different possible projections of a square that we get based on its orientation?

Q33

What do you think is the projection of a parallelogram under different orientations?

Can this ever be a quadrilateral that is not a parallelogram? As a starting point, you could think about the projection of a pair of parallel lines.

Q34

What can you say about the projection of an $n$ -sided regular polygon?

[Hint: Projection of a polygon is composed of the projections of its sides.]

Q35

How would the projections of a cube and a cone look?

Q36

See Figures 4.2–4.5. In each case, see if you can visualise another object that gives the same projection.

Q37

Find another object that makes the same projection as that of a given cone.

Q39

Have you played Tetris? There are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.

Q40

Context: In Fig. 4.8, there are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.

Q. Imagine these are cubes, not squares. Draw each of these on your isometric paper (you can find it at the end of the book).

Q41

Why is this correspondence between directions on isometric paper and axes of the solid so effective for communicating the shape of the solid?

Q42

Can you try drawing the other tetris shapes on isometric paper?