To find the side of the square, we need to understand the given information and use some geometric principles.

Given information:
- AB = 13
- BC = 21
- AC = 20

Let's solve this step by step:

1. Determine the type of triangle:
Looking at the given side lengths, we can determine the type of triangle ABC. By comparing the side lengths, we can see that BC is the longest side (21), which means it is opposite the largest angle. Therefore, triangle ABC is an obtuse triangle.

2. Draw a diagram:
We can draw triangle ABC with the given information. Let's draw triangle ABC with point D representing the vertex opposite to side BC. We can also draw a square inside the triangle, with one side lying on BC and the other three sides touching the sides of the triangle.

3. Find the area of triangle ABC:
Using the formula for the area of a triangle, we can calculate the area of triangle ABC. Let's use Heron's formula, which states that the area of a triangle with side lengths a, b, and c is given by:

Area = √(s(s-a)(s-b)(s-c))

where s is the semiperimeter of the triangle, given by:

s = (a + b + c) / 2

In this case, a = BC = 21, b = AC = 20, and c = AB = 13. Plugging these values into the formula, we get:

s = (21 + 20 + 13) / 2 = 54 / 2 = 27

Now, we can calculate the area of triangle ABC:

Area = √(27(27-21)(27-20)(27-13))
= √(27 * 6 * 7 * 14)
= √31752
≈ 178.22 (rounded to two decimal places)

4. Find the height of triangle ABC:
To find the height of triangle ABC, we can use the formula for the area of a triangle:

Area = (base * height) / 2

In this case, the base is BC = 21. Rearranging the formula, we can solve for the height:

height = (2 * Area) / base
= (2 * 178.22) / 21
≈ 16.96 (rounded to two decimal places)

5. Find the side of the square:
Since the side of the square lies on BC, the height of triangle ABC is equal to the side of the square. Therefore, the side of the square is approximately 16.96.

So, the side of the square is approximately 16.96.

The answer is 84/11
the above question is solved using the Heron's Formula
s=(13+21+20)/2 = 27

Area of triangle(A)= [27*(27-13)*(27-21)*(27-20)]^(.5) = 126

height of triangle(h)= 2*A / base = 2*126/21 = 12

from, similarity of triangles

h/(h-x) = BC / x where x is side of square
=> 12/(12-x) = 21/x
=> x = 84/11