Here is another animated solution: https://youtu.be/lEqQ7G-EtPw .  It solves a more general version of this question.

The key is to realize that in any equilateral triangle the 2 sides when combined have twice the length of the base. Once we understand that, it is immediately clear that the top part of the figure contributes 2 x AB and the bottom part 2 x AB, for a total of 4 x AB.

I think this method is more straightforward to visualize. Better for MCQs where no working is required and time should be optimized.

If the length of medium triangle is u cm, length of biggest triangle = 2* length of smallest triangle if you placed the medium triangle in mirror image from horizon.

length of biggest triangle = 21 – u

By inspection, perimter = 4u + 4(21 – u ) = 84 cm. 

Given that this is an MCQ question, I would do a quick drawing on paper to visually manipulate the line segments in order to solve this problem.

Here is an animated solution: https://youtu.be/a5urBc6nmB4  (it is in vertical YouTube shorts format so it might look better on a phone)

It solves a slightly more complex version of the original question but the idea is the same: no matter how many equilaterals triangles there are or whatever their sizes, the perimeter of the figure is always 4 times the width. (That’s why the solution in the original problem is 4 x 21 = 84.)

Here is another animated solution: https://youtu.be/lEqQ7G-EtPw .  It solves a more general version of this question.

The key is to realize that in any equilateral triangle the 2 sides when combined have twice the length of the base. Once we understand that, it is immediately clear that the top part of the figure contributes 2 x AB and the bottom part 2 x AB, for a total of 4 x AB.

I think this method is more straightforward to visualize. Better for MCQs where no working is required and time should be optimized.

If it is hard to imagine, then let the base of the 2 equilateral big triangles be a and b respectively.

Perimeter of equilateral Triangle A with base a = 3a

Perimeter of equilateral Triangle B with base b = 3b

a + b = XY = 21

Perimeter of 2 sides of Triangle A and 2 sides of Triangle B = 2(a + b) = 2 (21) = 42 

Now let the base of the other 3 equilateral triangles be c, d, e respectively

Perimeter of equilateral Triangle C with base c = 3c

Perimeter of equilateral Triangle D with base d = 3d

Perimeter of equilateral Triangle E with base e = 3e

c +d+ e = XY = 21

Perimeter of 2 sides of Triangle C and 2 sides of Triangle D and 2 sides of Triangle F  = 2(c + d + e)

                                                                                                                                                           = 2 (21)

                                                                                                                                                           = 42 

Hence Perimeter of the figure = 42 + 42 = 84cm

Hmm… interesting question.  It doesn’t matter how many triangles there are, if they are the equilateral triangles.  Just think of 2 big equilateral triangles of 21 cm.  Calculate 4 sides of the triangle so 21×4 = 84cm #