Find side AD in rectangle ABCD if diagonal AC = 13 cm and side AB = 12 cm.

The opposite sides of the rectangle are equal, so AD = BC.
Line AC divides rectangle ABCD into two right triangles ABC and ACD. To find the side BC, consider the triangle ABC.
In a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:
AB ^ 2 + BC ^ 2 = AC ^ 2;
BC = √ (AC ^ 2 – AB ^ 2) = √ (13 ^ 2 – 12 ^ 2) = √25 = 5.
AD = BC = 5 cm.
Answer: AD = 5 cm.