Find the area of a square if its diagonal is 3.

Given:
square ABCE,
AC – diagonal,
AC = 3.
Find the area of the square ABCE -?
Decision:
1. Consider a right-angled triangle ABC. By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AB ^ 2 + BC ^ 2 = AC ^ 2;
AB ^ 2 + BC ^ 2 = 3 ^ 2 (since all sides are equal in the square, AB = BC);
AB ^ 2 + AB ^ 2 = 9;
2 * AB ^ 2 = 9;
AB ^ 2 = 9: 2;
AB ^ 2 = 4.5.
2. Consider the square ABCE. Its area is its side squared.
Hence:
S ABCE = AB ^ 2.
Then S ABCE = 4.5.
Answer: 4.5.