In triangle ABC AB = 18, angle C is 45 degrees, find the radius of the circle circumscribed about triangle ABC

If the circle passes through all the vertices of the triangle, then it is circumscribed around it.

We find the radius (R) of such a circle, for which we use the extended theorem of sines, that is, the length of one side of the triangle (AB) is divided by the double sine of the opposite angle (∠С):

R = AB / 2 * sin ∠C.

Then R = 18/2 * sin 45 ° = 18 / (2 * √2: 2) = 18 / √2 = (18 * √2) / 2 = 9 * √2.

Answer: 9 * √2.