The Length Of Diagonal Of Rhombus Is 16 And 12

Agustus 14, 2023 Posting Komentar

The length of the diagonal of a rhombus are 16cm & 12cm. find the side from brainly.in

Are you a student struggling with geometry? Or maybe you are just curious about the length of the diagonal of a rhombus? Whatever the reason, you have found yourself here, and we are here to help. In this article, we will explore the length of the diagonal of a rhombus when the side lengths are given. So, let's dive in!

What is a Rhombus?

Before we get into the length of the diagonal, let's define what a rhombus is. A rhombus is a quadrilateral with all sides equal in length. It is a special case of a parallelogram where the opposite angles are congruent. Another important property of a rhombus is that its diagonals bisect each other at right angles.

Calculating the Length of the Diagonal

Now, let's get to the main topic of this article, which is calculating the length of the diagonal of a rhombus when the side lengths are given. In this case, we are given that the side lengths of the rhombus are 16 and 12. To calculate the length of the diagonal, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In a rhombus, the diagonals bisect each other at right angles, creating four right triangles with the diagonal as the hypotenuse.

Calculating the Longer Diagonal

Let's start by calculating the length of the longer diagonal. To do this, we need to find the length of one of the sides of the right triangle. We can use the fact that the diagonals bisect each other to find this length. Since the diagonals bisect each other, we can draw a perpendicular from one vertex of the rhombus to the longer diagonal, dividing it into two congruent segments.

Now, we have a right triangle where the hypotenuse is the longer diagonal, and one of the legs is 8 (half of the shorter side). Using the Pythagorean theorem, we get:

a² + b² = c²

8² + b² = 16²

64 + b² = 256

b² = 192

b = sqrt(192) = 8sqrt(3)

Therefore, the longer diagonal of the rhombus is:

c = 2b = 16sqrt(3)

Calculating the Shorter Diagonal

Now, let's move on to calculating the length of the shorter diagonal. To do this, we need to find the length of the other leg of the right triangle. We can use the fact that the diagonals bisect each other to find this length as well. Since the diagonals bisect each other, we can draw a perpendicular from one vertex of the rhombus to the shorter diagonal, dividing it into two congruent segments.

Now, we have a right triangle where the hypotenuse is the shorter diagonal, and one of the legs is 6 (half of the longer side). Using the Pythagorean theorem, we get:

a² + b² = c²

6² + b² = 12²

36 + b² = 144

b² = 108

b = sqrt(108) = 6sqrt(3)

Therefore, the shorter diagonal of the rhombus is:

c' = 2b = 12sqrt(3)

Conclusion

In conclusion, we have calculated the length of the diagonal of a rhombus when the side lengths are given. We used the Pythagorean theorem and the fact that the diagonals bisect each other to find the length of the longer and shorter diagonals of the rhombus. We hope this article has been helpful in your understanding of geometry and the properties of a rhombus.

Remember, practice makes perfect, so keep practicing and exploring the world of geometry!