You are given two squares, the diagonals of which are 12 and 13. Find the diagonal of the square

You are given two squares, the diagonals of which are 12 and 13. Find the diagonal of the square, the area of which is equal to the difference between the areas of these squares.

Find the length of the side of the square with the diagonal, equal to 12.

We denote it by x1.

Applying the Pythagorean theorem, we can compose the following equation:

x1 ^ 2 + x1 ^ 2 = 12 ^ 2,

solving which, we get:

2×1 ^ 2 = 144;

x1 ^ 2 = 144/2;

x1 ^ 2 = 72;

x1 = √72.

We find the length of the side of the square with the diagonal, equal to 13.

We denote it by x2.

Applying the Pythagorean theorem, we can compose the following equation:

x2 ^ 2 + x2 ^ 2 = 13 ^ 2,

solving which, we get:

2×2 ^ 2 = 169;

x2 ^ 2 = 169/2;

x2 = √ (169/2).

We find the difference in the areas of these squares:

x2 ^ 2 – x1 ^ 2 = 169/2 – 72 = 25/2.

Therefore, the area of ​​the required square is 25/2, and its side is √ (25/2).

Applying the Pythagorean theorem, we find the diagonal of the required square:

√ ((√ (25/2) ^ 2 + (√ (25/2) ^ 2) = √ (25/2 + 25/2) = √25 = 5.

Answer: 5.