In a right triangle, the hypotenuse c and the acute angle alpha * are known.

In a right triangle, the hypotenuse c and the acute angle alpha * are known. Find the legs, the area of the triangle and the radius of the circumscribed circle, if: c = 4, alpha = 30 *.

1) Find the BC leg.

Let us use the property that in a right-angled triangle the leg, which lies opposite an angle of 30 °, is equal to half the hypotenuse.

Hypotenuse AB = 4 (by condition), which means that leg BC = 2.

2) Find the AC leg.

Let’s use the Pythagorean theorem: the sum of the squares of the legs is equal to the square of the hypotenuse.

AB ^ 2 = AC ^ 2 + BC ^ 2.

Let us express AC ^ 2 from this formula: AC ^ 2 = AB ^ 2 – BC ^ 2.

AC ^ 2 = 4 ^ 2 – 2 ^ 2 = 16 – 4 = 12.

AC = √12 = √4×3 = 2√3.

3) Find the area of ​​the triangle ABC.

Let’s use the property: S of a right-angled triangle is equal to half the product of its legs.

S ABC = AC x BC / 2.

S ABC = 2√3 x 2/2 = 2√3.

4) Find the radius of the circumscribed circle.

Property: if a circle is described around a right-angled triangle, then its center is the midpoint of the hypotenuse.

It follows that the R of the circle is equal to half AB, that is, 2.

Answer: AC = 2√3; BC = 2; S = 2√3; R = 2.