In a triangle DEF, DE = 4cm, angle D = 30cm, angle E = 120cm. Find the unknown sides

In a triangle DEF, DE = 4cm, angle D = 30cm, angle E = 120cm. Find the unknown sides of the triangle and the radius of the circle around it.

By condition, in a triangle DE F angle E = 120, angle D = 30, then angle F = 180 – 120 – 30 = 30.

Then the triangle DEF is isosceles, DE = FE = 4 cm.

By the sine theorem, DE / Sin30 = DF / Sin120.

4 / (1/2) = DF / (√3 / 2).

DF = 8 * (√3 / 2) = 4 * √3.

The radius of a circle circumscribed around an isosceles triangle is determined by the formula:

R = DE ^ 2 / √ (4 * DE ^ 2 – DF ^ 2) = 4 ^ 2 / √ (4 * 4 ^ 2 – (4 * √3) ^ 2).

R = 16 / √ (64 – 48) = 16/4 = 4 cm.

Answer: The radius of the circumscribed circle is 4 cm.