Question

The perimeter of a rhombus is 146 cm and one of its diagonals is 48 cm, find the other diagonal and the area of the rhombus.

Answer 1

we know that,the diagonals of rhombus bisect each other at 90°.
the half of the diagonal =48×1/2=24cm
the each side=146/4=36.5cm
the haf of the other diagonal=√(36.5^2-24^2)=27.5cm
the length of the diagonal=27.5×2=55

Answer 2

${\huge {\underline {\bf {\blue {Question}}}}}$

The perimeter of a rhombus is 146 cm and one of its diagonals is 48 cm, find the other diagonal and the area of the rhombus.

${\huge {\underline {\bf {\blue {Answer}}}}}$

Given :-

• The perimeter of a rhombus = 146 cm
• One of its diagonal = 48 cm.

To Find :-

• Other diagonal of rhombus
• Area of the rhombus.

Solution :-

We know that, all sides of rhombus are equal.

${\boxed {\sf {Perimeter~ of~ rhombus = 4 \times side}}}$

$~$

Perimeter of rhombus = 146 cm

$~$

${\sf {4 \times side = 146~ cm}}$

$~$

${\sf {Side = {\dfrac {146}{4}} ~ cm}}$

$~$

${\sf {Side = 36.5~ cm}}$

$~$

Diagonals in a rhombus bisect each other at right angles. O is the midpoint of AC and BD. But BD = 55. therefore, OB = OD = 27.5 cm

In △AOB

∠AOB = 90°

By Pythagoras theorem,

$~$

$\implies$ ${\sf {{AB}^{2} = {AO}^{2} + {OB}^{2}}}$

$~$

$\implies$ ${\sf {{36.5}^{2} = {AO}^{2} + {27.5}^{2}}}$

$~$

$\implies$ ${\sf {{36.5}^{2} - {27.5}^{2} = {AO}^{2}}}$

We know,

${\boxed {\sf { (a + b) (a - b) = {a}^{2} - {b}^{2}}}}$

$~$

$\implies$ ${\sf { (36.5 + 27.5) (36.5 - 27.5) = {AO}^{2}}}$

$~$

$\implies$ ${\sf {64 \times 9 = {AO}^{2}}}$

$~$

$\implies$ ${\sf { {AO}^{2} = 576}}$

$~$

$\implies$ ${\sf {AO = 24 cm}}$

$~$

${\underline {\boxed {\sf { Length~ of~ diagonal~ AC = 24 + 24 = 48~ cm}}}}$

Also,

${\boxed {\sf { Area~ of~ Rhombus = {\dfrac {1}{2}} \times {d}_{1} \times {d}_{2} }}}$

$~$

$\implies$ ${\sf { Area~ of~ Rhombus = {\dfrac {1}{2}} \times 48 \times 55 }}$

$~$

$\implies$ ${\sf { Area~ of~ Rhombus = {\dfrac {1}{ {\cancel {2}}}} \times {\cancel {{48}}^{24}} \times 55 }}$

$~$

$\implies$ ${\sf { Area~ of~ Rhombus = 24 \times 55}}$

$~$

${\underline {\boxed {\sf { Area~ of~ Rhombus = 1320~ {cm}^{2}}}}}$

${\underline {\sf {\purple {Hence,~ Length~ of~ other~ diagonal~ is~ 48~ cm}}}}$

${\underline {\sf {\purple {Area~ of~ Rhombus~ is~ 1320 {cm}^{2}}}}}$