In triangle ABC, the outside angle at the vertex C is 90 degrees and the outside angle at the vertex A is 150 degrees.

In triangle ABC, the outside angle at the vertex C is 90 degrees and the outside angle at the vertex A is 150 degrees. the smaller side of a triangle is 12.5. Find the length of the diameter of the circle around this triangle.

Since the external angle of the ВСD, by condition, is equal to 90, then the internal angle ACB = 90, since the sum of adjacent angles is equal to 180.

Then the triangle ABC is rectangular.

The BAC angle is adjacent to the BAC angle, then the BAC angle = 180 – 150 = 30.

The BC leg lies against a smaller angle, therefore it is the smaller side of the triangle, then BC = 12.5 cm.

The BC leg lies opposite an angle of 30, then BC = AB / 2.

AB = 2 * BC = 2 * 12.5 = 25 cm.

The diameter of a circle circumscribed about a right-angled triangle is equal to the length of its hypotenuse. D = AB = 25 cm.

Answer: The diameter of the circle is 25 cm.