we conclude that, in a parallelogram, if all the sides are equal then it is called a rhombus.

**Rhombus**

Take four sticks of equal length and four connectors. Connect four
sticks to form a square as shown in the Fig.2.11(i). Then, try to make any two
opposite vertices closer as shown in the Fig.2.11(ii) such that opposite sides
remain parallel to each other to get a new shape called rhombus.

Hence, we conclude that, in a parallelogram, if
all the sides are equal then it is called a rhombus.

**Note**

In a rhombus, (i) all the sides
are equal (ii) opposite sides are parallel (iii) diagonals divide the rhombus
into 4 right angled triangles of equal area. (iv) the diagonals bisect each
other at right angles.

__1.
Area of the rhombus if base and height are given__

Draw a rhombus on a graph sheet as shown in the
Fig.2.12(i) and cut it. Draw a perpendicular line from one vertex to the
opposite side. Cut the triangle and shift the triangle to the other side of the
rhombus as shown in the Fig.2.12(ii).What shape do you see? It is a square.
Hence, the area of the rhombus is the same as that of the square.

Area of the square

= side ×
side *sq.units*

= base ×
height *sq.units*

= area
of the rhombus

** **

__2.
Area of the rhombus if the diagonals are given__

Let us find, area of the rhombus ABCD by
splitting it into two triangles.

Here AB = BC = CD = DA and diagonals AC (*d*_{1})
and BD (*d*_{2})
are perpendicular to each other.

Area of the rhombus ABCD = Area of triangle ABC
+ Area of triangle ADC

= 1/2 ×
AC × OB + 1/2 × AC × OD

= 1/2 ×
AC (OB+OD)

= 1/2 ×
AC × BD

= 1/2 × *d*_{1} × *d*_{2} *sq. units*

Therefore, area of the rhombus = 1/2 (product
of diagonals) *square units*.

** **

**Try these**

**1. Observe the Fig. 2.14 and
answer the following questions.**

**(i) Name two pairs of opposite
sides.**

Two pairs of opposite sides
PQ and RS, QR and PS.

**(ii) Name two pairs of adjacent
sides. **

Two pairs of adjacent sides
PQ and QR, PS and RS.

**(iii) Name the two diagonals **

Two diagonals PR and QS.

**2. Find the area of the rhombus
given in Fig. 2.15 and Fig. 2.16**

(i) d_{1} = 11 *cm,*
d_{2} = 13 *cm*

Area of the rhombus = ½ d_{1 }×_{ }d_{2}
*sq.units*

= 1/2 × 11 ×
13 *sq.cm*

= 143 / 2 =
**71.5
sq.cm**

(ii) base* b* = 10* cm*

height *h* = 7 *cm*

Area of the rhombus *= b × h
sq .units*

= 10 × 7 = **70 sq. cm**

__Example 2.6__

Find the area of the rhombus whose side is 17 *cm* and the height is 8 *cm*.

*Solution*

Given:

Base = 17 *cm*, height = 8 *cm*

Area of the rhombus = *b* × *h sq. units*

= 17 × 8 = 136

Therefore, area of the rhombus = 136 *sq. cm.*

**Think**

1. Can you find the perimeter of
the rhombus?

2. Can diagonals of a rhombus be
of the same length?

3. A square is a rhombus but a
rhombus is not a square. Why?

4. Can you draw a rhombus in such
a way that the side is equal to the diagonal.

__Example 2.7__

Calculate the area of the rhombus having
diagonals equal to 6 *m* and 8 *m*.

*Solution*

Given: *d*_{1} =
6 *m, d*_{2} =
8 *m*

Area of the rhombus = 1/2 × (*d*_{1} × *d*_{2}) *sq. units*

= 1/2 ×
(6 × 8)

= 48/2

= 24 *sq.m*

Hence, area of the rhombus is 24 *sq.m.*

__Example 2.8__

If the area of the rhombus is 60 *sq. cm* and one of the diagonals is 8 *cm*, find the length of the other diagonal.

*Solution*

Given, the length of one diagonal (*d*_{1})
= 8 *cm*

Let, the length of the other diagonal be *d*_{2}*
cm*

Area of the rhombus = 60 *sq. cm* (given)

1/2 × (*d*_{1}× *d*_{2}) = 60

1/2 × (8× *d*_{2})
= 60

8 × *d*_{2}
= 60 × 2

*d*_{2} = 120/8

= 15

Therefore, length of the other diagonal is 15 *cm.*

__Example 2.9__

The floor of an office building consists of 200
rhombus shaped tiles and each of its length of the diagonals are 40 *cm* and 25 *cm*. Find the total cost of polishing the floor at
₹ 45 per *sq.m.*

*Solution*

Given, the length of the diagonals of a rhombus
shaped tile are 40 *cm* and 25 *cm*

The area of one tile = 1/2 × (*d*_{1}×
*d _{2}* )

= 1/2 × 40 × 25

= 500 *sq. cm*

Therefore, the area of 200 such tiles = 200 ×
500

= 100000
*sq. cm*

= 100000/10000
(1 *sq. m =* 10000 *sq.
cm)*

= 10 *sq. m*

Therefore, the cost of polishing 200 such tiles

at the rate of ₹ 45 per *sq.
cm* = 10 × 45 = ₹ 450.

In railway the terminology,
“Diamond Crossing” refers to the point
where two railway lines cross, forming the shape of rhombus at the crossing
point. The most famous diamond crossing is at Nagpur, where lines from the
North, South, East, and Western railways meet.

Tags : Measurements | Term 1 Chapter 2 | 7th Maths , 7th Maths : Term 1 Unit 2 : Measurements

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