In a right-angled triangle ABC with a right angle C, the outer angle at the vertex A is 120

In a right-angled triangle ABC with a right angle C, the outer angle at the vertex A is 120 degrees, AC + AB = 18cm. Find AC and AB.

The condition says that in a right-angled triangle ABC, the right angle is C, and the outer angle at the vertex A is 120 °, AC + AB = 18 cm (the sum of the smaller leg and the hypotenuse). In order to find AC and AB find all the angles of the triangle.

So, the angle C is 90 °, and the outer angle at the vertex A is 120 °.

It is known from the property of the outer corner of a triangle that the outer angle is equal to the sum of two inner angles that are not adjacent to it.

Find angle B:

angle С = 90 °, so angle В = 120 – 90 = 30 °.

A leg lying opposite an angle of 30 ° is equal to half of the hypotenuse.

Smaller leg x cm, 2x – hypotenuse.

Let’s compose and solve the equation:

x + 2x = 18;

3x = 18;

x = 6.

So AC = 6 cm, and AB = 2 * 6 = 12 cm.