Solving for the Smallest Side of a Triangle Given Angle Ratios

In the realm of geometry, analyzing a triangle where the angles are in a specific ratio can lead to interesting insights and problem-solving exercises. Consider a triangle where the angles are in the ratio 1:2:3. If we know that the largest side of the triangle is 6 cm, how can we determine the length of the smallest side? Let's break down the problem step-by-step and explore the method to solve for the smallest side of the triangle.

Determine the Angles

Given that the angles of the triangle are in the ratio 1:2:3, we can denote these angles as x, 2x, and 3x. Since the sum of the angles in a triangle is 180°, we can write:

[x 2x 3x 180°]

[6x 180°]

[x 30°]

Therefore, the angles of the triangle are:

30° 60° 90°

This indicates that the triangle is a right triangle with angles of 30°, 60°, and 90°.

Identify the Sides

For a 30°-60°-90° triangle, the sides are in the ratio 1:√3:2, where:

The side opposite the 30° angle is the smallest side. The side opposite the 60° angle is the next side. The side opposite the 90° angle (the hypotenuse) is the longest side.

Let's denote the smallest side as a, the side opposite the 60° angle as a√3, and the hypotenuse as 2a.

Relate the Sides to the Given Largest Side

We know that the largest side, which is the hypotenuse, is given as 6 cm. Therefore:

[2a 6]

[a 3text{ cm}]

The smallest side of the triangle, which is opposite the 30° angle, can be determined as:

[a 3text{ cm}]

Why 30°-60°-90° Right Triangle?

Understanding why this is a 30°-60°-90° triangle is essential. The angles 30° and 60° are common in right triangles and their side ratios are well-known. This special triangle has proven to be very useful in trigonometry and geometry for solving various real-world problems.

Application in Real World

Understanding the properties of 30°-60°-90° triangles is particularly useful in fields such as architecture, engineering, and physics. These triangles frequently appear in the design of structures and in resolving vectors in physics. Knowing the 1:√3:2 ratio can help in quickly solving problems involving lengths and angles.

Conclusion

In summary, by identifying the angle ratios and applying the properties of 30°-60°-90° triangles, we can easily solve for the sides of a triangle given one side length. The length of the smallest side of the triangle, in this case, is 3 cm.

For further exploration, consider how trigonometric ratios like sine, cosine, and tangent relate to the sides of different triangles. Understanding these relationships can provide deeper insights into solving complex geometric problems.