# If the length of one side of a rhombus is 5cm and its diagonal is 8cm, find the length of the other diagonal.

Rhombus is also known as a four-sided quadrilateral. It is considered to be a special case of a parallelogram. A rhombus contains parallel opposite sides and equal opposite angles. A rhombus is also known by the name diamond or rhombus diamond. A rhombus contains all the sides of a rhombus as equal in length. Also, the diagonals of a rhombus bisect each other at right angles.

Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the ** Demo Class for First Step to Coding Course, **specifically **designed for students of class 8 to 12. **

The students will get to learn more about the world of programming in these **free classes** which will definitely help them in making a wise career choice in the future.

**Properties of a Rhombus**

A rhombus contains the following properties :

- A rhombus contains all equal sides.
- Diagonals of a rhombus bisect each other at right angles.
- The opposite sides of a rhombus are parallel in nature.
- The sum of two adjacent angles of a rhombus is equal to 180
^{o}. - There is no inscribing circle within a rhombus.
- There is no circumscribing circle around a rhombus.
- The diagonals of a rhombus lead to the formation of four right-angled triangles.
- These triangles are congruent to each other.
- Opposite angles of a rhombus are equal.
- When you connect the midpoint of the sides of a rhombus, a rectangle is formed.
- When the midpoints of half the diagonal are connected, another rhombus is formed.

**Diagonal of a Rhombus**

A rhombus has four edges joined by vertices. On connecting the opposite vertices of a rhombus, additional edges are formed, which result in the formation of diagonals of a rhombus. Therefore, a rhombus can have two diagonals each of which intersects at an angle of 90°.

**Properties of diagonal of a rhombus **

The diagonals of a rhombus have the following properties:

- The diagonals bisect each other at right angles.
- The diagonals of a rhombus divide into four congruent right-angled triangles.
- The diagonals of a rhombus may or may not be equal in length.

**Computation of diagonal of rhombus **

The length of the diagonals of the rhombus can be calculated by using the following methods:

**By Pythagoras Theorem **

Let us assume d_{1} to be the diagonal of the rhombus.

Since, we know, all adjacent sides in a rhombus subtend an angle of 90 degrees between them.

Therefore if we consider the rhombus ABCD,

In the triangle, BCD we have,

Since,

Perpendicular^{2 }+ Base^{2} = Hypotenuse^{2}

Putting the line segments,

BC^{2} + CD^{2} = BD^{2}

Now, we have,

In the case of a square rhombus, with all sides equal,

We know,

Square Diagonal: a√2where a is the length of the side of the square

In case of a rectangle rhombus, we have,

Rectangle Diagonal: √[l^{2}+ b^{2}]where,

- l is the length of the rectangle.
- b is the breadth of the rectangle.

### If the length of one side of a rhombus is 5cm and its diagonal is 8cm, find the length of the other diagonal.

**Solution:**

Since, we have,

Length of diagonal of the rhombus = 4 cm

Length of the side of the rhombus = 5 cm

We know, Diagonals of a rhombus bisect each other and they are perpendicular to each other.

Let us consider half of the length of second diagonal to be x.

Since, both the diagonals and a side form right angled triangle in case of a rhombus.

Therefore,

By Pythagoras theorem, we have,

Perpendicular

^{2 }+ Base^{2}= Hypotenuse^{2}⇒ x

^{2}+ 4^{2}= 5^{2}⇒ x

^{2}= 25 – 16⇒ x

^{2}= 9⇒ x = √9

⇒ x = 3

Thus, we get, length of second Diagonal d

_{2}= 2x⇒ 2 × 3 = 6 cm

Therefore,

The length of second diagonal d

_{2}is 6 cm.

### Sample Questions

**Question 1. The area of a rhombus is 10 cm sq. If the** **length of a diagonal is twice as long as the other diagonal. Then find the length of both the diagonals?**

**Solution:**

Assume,

The shorter diagonal be d

_{1}The longer diagonal be d

_{2}As it is given in the problem that the longer diagonal is twice the shorter diagonal

Thus this can be written as

d

_{2}= 2 × d_{1}……….equation (1)The area of the Rhombus is

A =

10 =

10 = ……..from equation (1)

10 =

d

_{1}= √10d

_{2}= 2 × √10d

_{2}= 2√10

**Question 2. Calculate the area of the rhombus that has each side 34 cm and the measure of its one diagonals is 32 cm.**

**Solution:**

Here assume ABCD is a rhombus

Thus,

34 cm = AB = BC = CD = AD

Diagonal BD = 32 cm

Here O is the diagonal intersection point

Thus,

BO = OD = 16 cm

Further.

In the ∆ AOD,

AD

^{2}= AO^{2}+ OD^{2}⇒ 34

^{2}= AO^{2}+ 16^{2}⇒ 1156 = AO

^{2}+ 256⇒ AO

^{2}= 1156 – 256⇒ AO

^{2}= 900⇒ AO = 30 cm

As,

AC = 2 × AO

= 2 × 30

AC = 60 cm

Area of rhombus =

=

= 960 cm

^{2}

**Question 3. Calculate the altitude of the rhombus having an area of 630 cm² and its perimeter is 360 cm.**

**Solution:**

Here it is given that

The perimeter of rhombus is 360 cm

Therefore,

The side of rhombus = = = 90 cm

Now,

The area of rhombus = base × height

⇒ 630 = 90 × h

⇒ h =

⇒ h =7 cm

Hence,

The altitude of the rhombus is 7 cm.

**Question 4. A hall of a building consists of 4000 rhombus-shaped tiles having diagonals 80 cm and 50 cm. Calculate the cost of buffing of all the tiles of the hall at the cost of ₹20 per square meter?**

**Solution:**

Here it is given that the length of the diagonals of each rhombus-shaped tile is 80 cm and 50 cm.

Thus,

The area of one tiles is; a =

a =

a = 2000 sq. cm

Changing the area is square metre

a =

a = 0.2 sq. metre

Further the area for 4000 tiles will be; = 4000 × 0.2 sq.m

= 800 sq.m

Cost of buffing 1 sq.m is ₹ 20

Cost of buffing 800 sq.m will be; 800 × 20

= ₹1600

**Question 5. Assume the diagonals of a rhombus-shaped glass are 15 cm and 24 cm. Calculate the area of the Rhombus glass?**

**Solution:**

We know,

Area of rhombus =

a =

a = 180 sq.cm

Therefore,

The area of the Rhombus shapes glass is 180 sq.cm.