Exploring the Geometry of a Rhombus: Calculating Side Lengths

A rhombus is a fascinating geometric shape with many unique properties, particularly with respect to its angles and diagonals. In this article, we will explore how to calculate the side length of a rhombus when given one of its interior angles and the length of its shorter diagonal.

Solution to the Problem

Consider a rhombus ABCD where one of its interior angles is 138 degrees, and its shorter diagonal AC is 38.6 meters. We need to find the length of its sides. Let's follow a step-by-step approach to solve this problem using geometric properties and trigonometry.

Step 1: Setting Up the Rhombus and Its Diagonals

First, let's denote the vertices of the rhombus as A, B, C, and D. The diagonals AC and BD intersect at point O.

Given:

DAB DCB 138°

Diagonal AC 38.6 m

Diagonals of a rhombus bisect each other at right angles, so:

AO CO 19.3 m (half of the shorter diagonal) AOB 90° OAB 69° (since 180° - 2 times; 138°/2 69°)

Step 2: Applying Trigonometric Properties

Now that we have the angle and the length of AO, we can use trigonometric ratios to find the side length (AB) of the rhombus.

Consider the right-angled triangle AOB:

Solution 1: Using Cosine

Using the cosine function: [cos(69°) frac{AO}{AB}] [0.3583679495453 frac{19.3}{AB}] [AB frac{19.3}{0.3583679495453} approx 53.86 , text{m}]

Solution 2: Using the Sine Rule

Using the sine rule in triangle AOB: [frac{AB}{sin(90°)} frac{19.3}{sin(21°)}] [AB frac{19.3}{sin(21°)} approx frac{19.3}{0.35836795} approx 53.86 , text{m}]

Conclusion

Thus, the length of a side of the rhombus is approximately 53.86 meters. This method can be generalized to solve similar problems involving rhombi with one given interior angle and the length of the shorter diagonal.

Key Concepts and Interrelated Mathematical Concepts

This problem involves several important mathematical concepts, including:

Rhombus Properties: Diagonals bisect each other at right angles and bisect the angles. Trigonometry: Application of sine and cosine functions in right-angled triangles. Geometric Reasoning: Understanding and applying the properties of geometric shapes.

Related Keywords

Rhombus Diagonal Interior Angle

Further Exploration

For a deeper understanding of geometric shapes and trigonometry, you might want to explore:

Relationships between different angles in a rhombus Calculating the area of a rhombus using diagonals Properties of diagonals in various quadrilaterals

By understanding these concepts, you can apply them to solve a wide range of geometric problems and deepen your knowledge of mathematics.