## What Are Diagonals of a Rectangle?

**Diagonals of a rectangle are line segments connecting opposite vertices of a rectangle. **A rectangle is a quadrilateral in which opposite sides are equal and all angles measure $90^\circ$.

A diagonal is a line segment connecting two non-adjacent vertices of a polygon. The word diagonal comes from the ancient Greek word “*diagonios*,” (Latin *diagonalis)** *which means “from angle to angle.”

Both Euclid and Strabo used it to describe a line that connects two vertices of a cuboid or a rhombus; later, it became known in Latin as diagonus (slanting line).

### Diagonals of a Rectangle Definition

**A line segment that connects any two opposite vertices of a rectangle is called diagonal of a rectangle.**

Consider ▭ABCD. A rectangle with diagonals AC and BD as shown below.

Here, $AC = BD$.

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## Properties of Diagonals of a Rectangle

- The diagonals of a rectangle are congruent.

$\left[AC = BD\right]$

- Each diagonal divides a rectangle into two congruent right-angled triangles.

- The diagonals of a rectangle are
**not perpendicular**to each other.

- If two diagonals of a rectangle bisect each other at $90^\circ$, it is called a square.

- When two diagonals intersect, they form one obtuse angle and one acute angle. The opposite central angles are equal.

- Adjacent angles formed by the diagonals are supplementary.

- Diagonals of a rectangle form the hypotenuse of the right triangle.

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## Diagonal of a Rectangle Formula

The diagonal of a rectangle formula helps in finding the length of the diagonal when the length and width of the rectangle are known.

A diagonal divides a rectangle into two right triangles, each of which has a hypotenuse and sides that are equal to the sides of the rectangle. The diagonal is that hypotenuse.

So, how to find the diagonal of a rectangle? What is the diagonal measurement of a rectangle?

The diagonal length of a rectangle is calculated using the formula, $d = \sqrt{(w^2 + l^2)}$

where, $d = diagonal$

$l = length$

$w = width$

This formula can be used to find the measurement of the length of a diagonal of a rectangle.

## Diagonal of a Rectangle Formula Derivation

The diagonal of a rectangle formula is derived using the Pythagoras’ Theorem. Split the rectangle into two right triangles formed by a diagonal.

Applying Pythagoras’ Theorem in the right triangle PSR, we have

$d^2 = b^2 + l^2$

Here, $d =$ diagonal, $l =$ length, $b =$ breadth

Taking square root on both sides,

$d = \sqrt{(b^2 + l^2)}$

- $l =$ length of the rectangle
- $b =$ breadth of the rectangle

Simply substitute the values of the length and breadth in the formula to get the answer.

Example:

In the above rectangle,

Length of diagonal $= d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ units

## Angles Made by Diagonals of a Rectangle

As discussed above, if the diagonals of the polygon bisect each other, it is called a square.

Diagonals of a rectangle are equal in length, bisect each other. They do not meet at a right angle in the center. The diagonals of a rectangle do not bisect the interior angles of the rectangle.

**Example: **Take a look at the image given below.

In the image given above, AC and BD are the diagonals.

Angles formed by diagonal AC and BD are $\angle AED,\; \angle AEB,\; \angle BEC$, and $\angle DEC$.

These angles do not meet at right angles, but the adjacent angles are supplementary, which means the adjacent angle-pairs add up to $180^\circ$.

- $\angle AED + \angle AEB = 180^\circ$
- $\angle AEB + \angle BEC = 180^\circ$
- $\angle BEC + \angle DEC = 180^\circ$
- $\angle AED + \angle DEC = 180^\circ$

## Facts about Diagonals of a Rectangle

- The two diagonals of a rectangle divide the rectangle into four triangles.

- Each diagonal of a rectangle divides it into two congruent right triangles.

- The intersection of the diagonals is the circumcenter. That is, you can draw a circle with that at the center to pass through the four corners.

Thus, diagonal $= 2r$ …where r is the radius of the circumcircle

- We can calculate the length of the diagonal of the rectangle using the formula:

$d = \sqrt{(b^2 + l^2)}$

- The rectangle is called a square if its diagonals bisect each other at right angles.

- As the diagonals are generally slanted, a slanted symbol “ / ” is used to denote diagonals.

## Conclusion

In this article, we have learned about diagonals of the rectangle, its properties, formulas, and facts. Now let’s solve some examples based on the formulas learned above.

## Solved Examples for Diagonals of a Rectangle

**1. Identify the length, width and diagonal in the given rectangle. **

**Solution: **

Length $\rightarrow$ AD and BC

Width $\rightarrow$ AB and CD

Diagonals $\rightarrow$ AC and BD

**2. Find the length of each diagonal of a rectangle of length 15 units and width 8 units.**

**Solution: **

Length of the rectangle, $l = 15$ units.

Breadth of the rectangle, $b = 8$ units.

Length of the diagonal $= d = \sqrt{(b^2 + l^2)}$

$d = \sqrt{(8^2 + 15^2)}$

$d = \sqrt{(64 + 225)}$

$d = \sqrt{289}$

$d = 17$ units.

The length of the diagonal is 17 units.

**3. A rectangular park is 30 ft long and 16 ft wide. Determine the diagonal of the rectangular park.**

**Solution:**

Length of the rectangular park $= 30$ ft

Breadth of the rectangular park $= 15$ ft

Length of the diagonal $d = \sqrt{(b^2 + l^2)}$

$= sqrt{(16^2 + 30^2)}$

$= \sqrt{(256+ 900)}$

$= \sqrt{1156}$

$= 34$ ft

The diagonal of the rectangular park is 34 ft.

**4. Find the length of the diagonal of the given rectangle.**

**Solution: **

Length $= 12$ in

Breadth $= 5$ in

Diagonal $= ?$

Using the Pythagorean theorem, we get:

$l^2 + b^2 = d^2$

$12^2 + 5^2 = d^2$

$d^2 = 144 + 25$

$d^2 = 169$

Taking the square root

$d = 13$ in

**5. Find the length of the rectangle if the breadth of the rectangle is 3 inches, and the diagonal of the rectangle is 5 inches.**

** Solution: **

Breadth of the rectangle $= 3$ in

Diagonal of the rectangle (hypotenuse) $= 5$ in

Here we will apply Pythagoras’ Theorem.

$(Hypotenuse)^2 = (Length)^2 + (Breadth)^2$

$5^2 = l^2 + 3^2$

$l^2 = 5^2 \;-\; 3^2$

$l^2 = 25 \;-\; 9$

$l = \sqrt{16}$

$l = 4$ in

Length of the rectangle is 4 inches.

## Practice Problems on Diagonals of a Rectangle

## Diagonal of a Rectangle - Properties, Formula, Examples, FAQs

### A line segment that connects any two opposite vertices of a rectangle is said to be its _________.

A line segment that connects any two opposite vertices of a rectangle is said to be its diagonal.

### The diagonals of a rectangle are ______.

The diagonals of a rectangle are equal in length.

### Which of the following is NOT TRUE ?

The diagonals of a rectangle are not perpendicular to each other. When two diagonals bisect each other at $90^\circ$, it is called a square.

### The length of diagonals of the rectangle is calculated using formula _________.

The length of diagonals of the rectangle is calculated using formula: $d = \sqrt{(b^2 + l^2)}$

where, $d =$ diagonal of rectangle

$l =$ length of the rectangle

$b =$ breadth of the rectangle

### Adjacent angles formed by the diagonals of a rectangle are ________.

Adjacent angles formed by the diagonals are supplementary.

## Frequently Asked Questions about Diagonals of a Rectangle

**Why are the diagonals of a rectangle equal in length?**

A rectangle is a quadrilateral where all the angles are right angles. A rectangle is also a parallelogram where the opposite sides are equal.

Consider this rectangle ABCD.

In the above rectangle, consider triangles ABC and DCB.

$\angle ABC = \angle DCB = 90$ [Angles of rectangle]

$BC = BC$ (common side)

$AB = DC$ (Opposite sides of a parallelogram are equal)

Hence, $ABC\congDCB$

$AC = DB$ [CPCTC]

Hence, the diagonals of a rectangle are equal.

**Do diagonals of a rectangle always bisect each other?**

The diagonals of a rectangle always bisect each other. However, they do not meet at 90 degrees except if the rectangle is a square.

**What is the formula for the area of a rectangle?**

Area of a rectangle is given by the product of length and breadth.Area of rectangle $= l\timesb$

**Why is the area of the rectangle the product of length and breadth?**

Let us understand why the area of the rectangle is $length \times breadth$.

Consider ▭ ABCD. We can see that the diagonal AC divides the rectangle into two congruent triangles, $\Delta ABC$ and $\Delta ADC$.

Area of Rectangle $ABCD =$ Area of Triangle $ABC +$ Area of Triangle ADC

$= 2\times$ Area of Triangle ABC

$= 2 \times (\frac{1}{2} \times Base \times Height)$

$= AB \times BC$

$= Length \times Breadth$

The area of the rectangle is the product of length and breadth.

**How many diagonals are there in a rectangle?** **Are the diagonals of a rectangle always congruent?**

A rectangle has two diagonals. Each diagonal divides a rectangle into two right triangles. The two diagonals of a rectangle are always congruent.

**Are diagonals perpendicular in a rectangle?**

No! The two diagonals of a rectangle do not intersect at right angles.