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

In a right-angled triangle ABC with a right angle C, the outer angle at the apex A is 120 degrees, hypotenuse AB = 13 cm Find the leg AC.

Since, according to the condition of this problem, we know the length of the hypotenuse and the value of the external angle at the vertex A, then to calculate the length of the leg AC we use the cosine theorem, according to which the cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:

cos A = AC / AB;

AC = AB cos A.

To do this, we find the degree measure of angle A. Since the sum of the outer and inner angles at one vertex is 180º, and the outer angle at the vertex A is 120º, then:

∠А = 180º – φ;

∠А = 180º – 120º = 60º;

cos 60º = 1/2 = 0.5;

AC = 13 0.5 = 6.5 cm.

Answer: the length of the AC leg is 6.5 cm.