Finding the Length of the Other Diagonal in a Rhombus
A rhombus is a type of quadrilateral with all sides of equal length. Like other quadrilaterals, a rhombus also has diagonals that intersect at right angles and bisect each other. In this article, we will explore how to calculate the length of the other diagonal of a rhombus given the length of one side and one diagonal.
Explanation and Solution
Let us consider a rhombus where one side (x) is 13 cm and one of its diagonals (d1) is 10 cm. We are required to find the length of the other diagonal (d2).
Using the Properties of a Rhombus
According to the properties of a rhombus, the diagonals bisect each other at right angles. This means we can split the rhombus into four right-angled triangles, each with one side of the rhombus as the hypotenuse and half the lengths of the diagonals as the other two sides.
Formula Application
The relationship between the sides and diagonals of a rhombus can be given by the formula:
d1^2 d2^2 4x^2
Substituting the values, we get:
Step 1: Substitute given values in the formula:
10^2 d2^2 4(13^2)
Step 2: Simplify the equation:
100 d2^2 4(169)
Step 3: Calculate the value:
100 d2^2 676
Step 4: Isolate d2^2:
d2^2 676 - 100
d2^2 576
Step 5: Solve for d2:
d2 √576 cm
d2 24 cm
The length of the other diagonal is 24 cm.
Alternative Method Using Right Triangles
Each diagonal bisects the rhombus into two congruent right-angled triangles. Let’s call one of these diagonals d1 and the other d2. The half-diagonals form the legs of the triangle, and the side of the rhombus is the hypotenuse.
Using Pythagoras Theorem
Considering one of these right-angled triangles, with the sides being half-diagonal1 (5 cm), half-diagonal2 (d/2), and the side of the rhombus (13 cm), we can use the Pythagorean theorem to find the length of the other half-diagonal (d/2).
Step 1: Set up the equation:
(d/2)^2 13^2 - 5^2
Step 2: Calculate the value:
(d/2)^2 169 - 25
(d/2)^2 144
Step 3: Solve for d/2:
d/2 √144
d/2 12 cm
Step 4: Determine d2:
d2 2(d/2) 2(12) 24 cm
The length of the other diagonal is 24 cm.
Conclusion
By using either the given formula or by breaking down the problem into smaller right-angled triangles, we can find the length of the other diagonal of a rhombus. In our example, the length of the other diagonal is 24 cm. Additionally, we can calculate the area of the rhombus using the formula:
Area (d1 * d2) / 2
In this example, the area would be:
Area (10 * 24) / 2 120 square cm
This method not only helps in finding the dimensions but also in calculating the properties like the area of the rhombus.
Additional Insights on Rhombus Properties
Rhombi are rich in properties, including the fact that their diagonals not only bisect each other but also divide the rhombus into four right-angled triangles. These properties can be applied to a wide range of geometry problems, making understanding the properties of a rhombus a valuable skill in both academic and practical contexts.