Introduction

Determining the side length of a rhombus, given the lengths of its diagonals, is a classic problem that involves the application of the Pythagorean theorem. This article will guide you through the process step-by-step, providing a clear and concise explanation alongside the relevant formulae. By the end of this post, you will have a solid understanding of how to solve such problems.

Understanding the Problem

A rhombus is a quadrilateral where all four sides are of equal length. One of the key properties of a rhombus is that its diagonals bisect each other at right angles (90 degrees). This property can be leveraged to find the side length of the rhombus when the lengths of the diagonals are known.

Step-by-Step Solution

To find the length of the side of a rhombus given the lengths of its diagonals, follow these steps:

Identify the diagonals: Let (d_1 16 , text{cm}) and (d_2 12 , text{cm}). Find the lengths of the half-diagonals: The diagonals bisect each other, so: (frac{d_1}{2} frac{16}{2} 8 , text{cm}) (frac{d_2}{2} frac{12}{2} 6 , text{cm}) Use the Pythagorean theorem: The side of the rhombus can be calculated using the right triangle formed by the half-diagonals.

The side length (s) of the rhombus can be calculated using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (side of the rhombus in this case) is equal to the sum of the squares of the other two sides (half-diagonals). (s^2 left(frac{d_1}{2}right)^2 left(frac{d_2}{2}right)^2 8^2 6^2 64 36 100)

Hence, the length of the side of the rhombus is:

(s sqrt{100} 10 , text{cm})

Example and Verification

Consider a rhombus where the diagonals are 16 cm and 12 cm. Let's find the length of its side:

Diagonals: (d_1 16, text{cm}) and (d_2 12, text{cm}). Half-diagonals: (frac{d_1}{2} 8, text{cm}) and (frac{d_2}{2} 6, text{cm}). Side length calculation: Using the Pythagorean theorem, (s sqrt{8^2 6^2} sqrt{100} 10, text{cm}).

This method can be verified by checking the consistency of the result.

Key Concepts

For a rhombus, the diagonals intersect at right angles and bisect each other. This property forms a right triangle, where the sides of the rhombus are the hypotenuses. The Pythagorean theorem is a powerful tool to find the length of the side of the rhombus.

Conclusion

To summarize, the length of the side of a rhombus can be found using the diagonals by leveraging the Pythagorean theorem. Understanding the properties of a rhombus and the application of the theorem allows for an efficient solution to such problems.