The bisector of a right-angled isosceles triangle drawn to the base is 3 cm. Find the sides of the triangle.

Given a right-angled isosceles triangle ACB with a right angle C. Draw the bisector CM = 3 cm to the hypotenuse. Since the triangle is isosceles, the bisector CM is height, therefore triangles CAM and CMB are rectangular. From the CAM triangle, we find the hypotenuse AC by determining the cosine of the ACM angle: cos ACM = MC / AC, hence AC = MC / cos ACM = 3 / cos 45 = 3 / (√2 / 2) = 6 / √2 = (6 * √ 2) / (√2 * √2) = 6√2 / 2 = 3√2 (cm). Leg CB = AC = 3√2 (cm).

By the Pythagorean theorem AB = √ ((3√2) ^ 2 + (3√2) ^ 2) = √ (18 + 18) = √ 36 = 6 (cm).

Answer: 3√2, 3√2, 6.