One of the corners of a right-angled triangle is 60 degrees, and the sum of the hypotenuse

One of the corners of a right-angled triangle is 60 degrees, and the sum of the hypotenuse and the smaller leg is 42 cm. Find the hypotenuse and the smaller leg.

1. In a right-angled triangle ABC, the sum of the acute angles is 90 °:

∠A = 60 °;
∠B = 90 ° – ∠A = 90 ° – 60 ° = 30 °.
2. The leg AC, opposite an angle of 30 °, is equal to half of the hypotenuse:

AC = 1/2 * AB, from here we get:
AB = 2 * AC.
3. Let’s compose and solve the equation for the second condition of the problem:

AB + AC = 42;
2 * AC + AC = 42;
3 * AC = 42;
AC = 42: 3;
AC = 14 (cm);
AB = 2 * AC = 2 * 14 = 28 (cm).
Answer: 14 cm and 28 cm.