Question

The measure of the largest angle of a certain triangle is 3 times the measure of the smallest angle. The measure of the remaining angle of the triangle is the average of the measures of the largest and smallest angles.

Write and solve the system of equations that describes the measure of each angle of the triangle.

Answer 1

The smallest angle of the triangle is 30 degrees, the largest is 90 degrees, and the remaining angle is 60 degrees, with the triangle being a 30-60-90 right triangle.

Step-by-step explanation:

Let's denote the smallest angle as x, then the largest angle is 3x, and the remaining angle, being the average of the largest and smallest, is ½(x + 3x) = 2x. The sum of the angles in a triangle is always 180 degrees.

Therefore, the sum of these three angles must be equal to 180 degrees.

The system of equations based on the given information is:

• x + 3x + 2x = 180

Simplifying this, we get:

• 6x = 180

Dividing both sides by 6, we find:

• x = 30

Thus, the smallest angle measures 30 degrees, the largest angle measures 3 × 30 = 90 degrees, and the remaining angle measures 2 × 30 = 60 degrees.

This triangle has angles of 30, 60, and 90 degrees, which are also the measures of angles in a 30-60-90 right triangle, a special type of triangle with its own set of trigonometric ratios.