In triangle ABC, angle B is 45 degrees, AC = 4 √2 cm. Find the diameter of the circle circumscribed about the triangle.

Let’s draw the diameter of the circle AD through point A, and connect the center of the circle, point O to the vertex C of the triangle ABC.

The inscribed angle ABC is equal to 45 and is based on the arc AC, the degree measure of which is equal to two meths of this angle. Arc AC = 2 * 45 = 90.

Then the value of the central angle AOC is equal to the magnitude of the arc AC. Angle AOC = 90.

In a right-angled triangle AOS, legs OA = OC = R.

Then, by the Pythagorean theorem, 2 * AO ^ 2 = AC ^ 2 = 32.

AO ^ 2 = 16.

AO = R = 4 cm.

Then D = АD = 2 * R = 2 * 4 = 8 cm.

Answer: The diameter of the circle is 8 cm.