Are Diagonals Of Rhombus Equal?

November 09, 2023 Posting Komentar

A rhombus ABCD has AB = 10 and m∠A = 60°. Find the lengths of the from brainly.in

When it comes to geometry, rhombus is one of the most interesting shapes. It is a quadrilateral with all sides of equal length, and opposite sides parallel to each other. However, what makes rhombus unique is its diagonals. The diagonals of rhombus have some special properties that make them stand out from other quadrilaterals. In this article, we will explore the question, "Are diagonals of rhombus equal?" Let's dive in!

What is a Rhombus?

A rhombus is a four-sided polygon with all sides of equal length. It is also known as a diamond or a lozenge. Rhombus has opposite sides parallel to each other, and its angles are not necessarily right angles. In other words, rhombus is a special type of parallelogram with all sides equal.

Properties of a Rhombus

Here are some important properties of a rhombus:

All sides are equal in length.
Opposite sides are parallel to each other.
Diagonals are perpendicular to each other.
Diagonals bisect each other.
The sum of the squares of the diagonals is equal to four times the square of the side length.

Are Diagonals of Rhombus Equal?

Yes, the diagonals of rhombus are always equal in length. This is a unique property of rhombus, and it can be proved mathematically.

Proof:

Let's consider a rhombus ABCD, where AB = BC = CD = DA. Let the diagonals intersect at point O. We need to prove that AO = BO = CO = DO.

First, we know that diagonals bisect each other, so AO = CO and BO = DO.

Now, let's draw perpendiculars from point O to sides AB and CD. Let the points of intersection be E and F, respectively.

Since ABCD is a rhombus, we know that AB is parallel to CD. Therefore, OE is perpendicular to AB, and OF is perpendicular to CD.

Also, we know that AB = CD, and AE = EC (because OE is the perpendicular bisector of AB). Similarly, BF = FD (because OF is the perpendicular bisector of CD).

Now, we can use the Pythagorean theorem to find the length of each diagonal:

AO² = AE² + OE² (by Pythagoras theorem)

CO² = CE² + OE² (by Pythagoras theorem)

BO² = BF² + OF² (by Pythagoras theorem)

DO² = DF² + OF² (by Pythagoras theorem)

Since AE = CE and BF = DF, we can simplify the above equations:

AO² = CO²

BO² = DO²

Therefore, AO = BO = CO = DO. Hence, the diagonals of rhombus are always equal in length.

Why are Diagonals of Rhombus Equal?

The reason why the diagonals of rhombus are always equal is because of the symmetry of the shape. Since rhombus has all sides of equal length, its opposite angles are also equal. When we draw the diagonals, they intersect at the center of the rhombus, which is equidistant from all four sides. This creates a symmetrical shape, where all diagonals are equal in length.

Applications of Rhombus

Rhombus has many applications in real life, such as:

Rhombus-shaped tiles can be used to create interesting patterns on floors and walls.
Rhombus-shaped mirrors can be used in optical instruments to reflect light at different angles.
Rhombus-shaped sails can be used in sailboats to catch the wind from different directions.

Conclusion

In conclusion, the diagonals of rhombus are always equal in length. This is a unique property of rhombus that makes it stand out from other quadrilaterals. The symmetry of the shape is the reason why all diagonals are equal. Rhombus has many applications in real life, and its properties are useful in various fields such as geometry, architecture, and engineering. Understanding the properties of rhombus can help us appreciate the beauty and complexity of geometry.

So, the next time someone asks you, "Are diagonals of rhombus equal?" You can confidently answer, "Yes, they are!"