Finding the Length of a Rhombus's Longer Diagonal Using Trigonometry and Geometry

Understanding the properties of a rhombus and applying trigonometry can help solve various geometric problems. In this article, we will explore how to find the length of a rhombus's longer diagonal when one angle and the side length are given. We will also discuss the geometric properties of the rhombus and how they relate to the problem at hand.

Properties of a Rhombus

A rhombus is a special type of parallelogram where all four sides are of equal length. Here are some key properties of a rhombus:

Diagonals of a Rhombus

Act as angle bisectors: The diagonals of a rhombus bisect the angles at the vertices. Perpendicular bisectors: The diagonals are perpendicular to each other and bisect each other.

These properties make the diagonals of a rhombus particularly useful in solving geometric problems. In the problem at hand, we are given that one of the angles of a rhombus is 60°, and each side of the rhombus is 16 cm. We need to find the length of the longer diagonal.

Step-by-Step Solution

Let's denote the rhombus as ABCD with each side of length 16 cm. Assume that one of the angles, say ∠A, is 60°. The longer diagonal is denoted as AC and the shorter diagonal as BD. The diagonals intersect at point O, which is the midpoint of both diagonals.

Step 1: Forming Equilateral Triangles

Since ∠A 60° and each side of the rhombus is 16 cm, we can see that triangle ABD is an equilateral triangle. Therefore, the length of the shorter diagonal BD is 16 cm, and since the diagonals bisect each other at right angles, BO OD 8 cm.

Step 2: Using Trigonometry to Find the Longer Diagonal

Now, consider triangle AOB, which is a right-angled triangle at O. Using the Pythagorean theorem:

AB2 AO2 BO2

or

162 AO2 82

128 AO2 64

AO2 64

AO 8√3 cm

Since AC 2AO, the length of the longer diagonal AC is:

AC 2 × 8√3 16√3 cm

Alternate Method Using Properties of a Rhombus

Another method to find the length of the longer diagonal involves recognizing that the diagonals of a rhombus split the rhombus into four congruent right triangles. The hypotenuse of these triangles is the side of the rhombus, and the angles are 30° and 60°. Let's solve this using the properties of a 30°-60°-90° triangle.

Step 1: Determine the Sides of a 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is (frac{sqrt{3}}{2}) times the hypotenuse.

Given that the side of the rhombus (the hypotenuse of the 30°-60°-90° triangle) is 16 cm, the side opposite the 60° angle is:

16 × (frac{sqrt{3}}{2}) 8√3 cm

This side is half of the longer diagonal. Therefore, the length of the longer diagonal is:

16√3 cm

Conclusion

By understanding the properties of a rhombus and applying trigonometric principles, we can determine the length of the longer diagonal. The longer diagonal of the rhombus is 16√3 cm when one of the angles is 60° and the side length is 16 cm. This problem demonstrates the importance of recognizing geometric properties and using appropriate mathematical tools to solve complex problems.