In a right-angled triangle ABC C = 90 B = 45 AC = 4√2. Find the length of the median CM and the area of the triangle ABC.

Since the triangle ABC is rectangular, and the angle ABC = 450, this triangle is isosceles, AC = AB.

Then, by the Pythagorean theorem, AB ^ 2 = AC ^ 2 + BC ^ 2 = 2 * AC ^ 2.

2 * AC ^ 2 = 32.

AC ^ 2 = 32/2 = 16.

AC = BC = 4 cm.

Determine the area of the triangle ABC.

Savs = AC * BC / 2 = 4 * 4/2 = 8 cm2.

Since triangle ABC is isosceles, the median CM is also the height of triangle ABC.

Then Saws = CM * AB / 2.

CM = 2 * Savs / AB = 2 * 8/4 * √2 = 4 / √2 = 2 * √2 cm.

Answer: The area of the triangle is 8 cm2, the length of the median is 2 * √2 cm.