How will the area of a square change if its diagonal is increased by 4 times?

The diagonal of a square can be determined using the Pythagorean theorem, since the diagonal divides the square into two isosceles right triangles. If we designate the diagonal as “c” and the sides of the square as “a”, the diagonal will be:

c ^ 2 = a ^ 2 + a ^ 2 = 2 * a ^ 2;

c = a * √‾2;

a ^ 2 = c2 / 2;

In turn, remember that the area of ​​a square corresponds to the product of two sides or the square of one of the sides of the square (all sides in the square are equal). Also, taking into account the calculations made, the area can be expressed through the diagonal:

S = a * a = a ^ 2 = c ^ 2/2;

If the diagonal increases by 4 times, then you can determine the new length of the side of the square:

C = 4 * c = 4 * a * √‾2.

C2 = (4 * c) ^ 2 = (4 * a * √‾2) ^ 2 = 16 * a ^ 2 * 2 = 32 * a ^ 2;

S = C ^ 2/2 = 32 * a ^ 2/2 = 16 * a ^ 2;

Thus, the area of ​​the square with the diagonal increased by 4 times is 16 times larger than the original.