# Given: Triangle ABC – rectangular Angle B = 60 degrees. The sum of the hypotenuse and the smaller

Given: Triangle ABC – rectangular Angle B = 60 degrees. The sum of the hypotenuse and the smaller leg = 18 cm.Find what is equal to hypotenuse and smaller leg.

First, let’s find what the second acute angle of this right-angled triangle is equal to.
According to the condition of the problem, the value of the angle B is 60 °.
Let angle A be right. Then the value of the angle C is:
∠С = 180 – ∠А – ∠В = 180 – 90 – 60 = 90 – 60 = 30 °.
Since in a triangle the smaller side always lies opposite the smaller angle of this triangle, in this triangle the smaller leg is the AB side, and the hypotenuse is the BC side.
Applying the sine theorem, we get:
| AB | = | BC | * sin (30 °) / sin (90 °) = | BC | / 2.
According to the condition of the problem, the sum of the hypotenuse and the smaller leg of this triangle is 18 cm, therefore, the following relation holds:
| BC | + | BC | / 2 = 18,
whence follows:
3 * | BC | / 2 = 18;
| BC | = 18 * 3/2;
| BC | = 27 cm.
Find | AB |:
| AB | = 27/2 = 13.5 cm.
Answer: the hypotenuse is 27 cm, and the smaller leg is 13.5 cm.

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