The angles of a triangle are 1: 2: 3. Find the ratio of the longest side of the triangle to its smallest side.

Let us denote by a the value of the largest angle of this triangle.

In the initial data for this task, it is reported that the values ​​of the angles of this geometric figure are related as one to two to three, which is the same, therefore, the values ​​of the other two angles of this triangle should be 3a and 2a degrees.

Since the sum of the values ​​of all 3 angles of a triangle is always 180 °, we can make the following equation:

a + 2a + 3a = 180,

solving which, we get:

6a = 180;

a = 180/6 = 30.

Therefore, the angles of this triangle are 30 °, 2a = 2 * 30 = 60 ° and 3a = 3 * 30 = 90 °.

The longest side of this triangle is opposite a 90 ° angle, and the smallest is opposite a 30 ° angle.

To find the ratio of the largest side c to the smallest side a, we use the theorem of sines:

s / a = sin (90 °) / sin (30 °) = 1 / (1/2) = 2.

Answer: the desired ratio is 2: 1.