Question

if the diagonal and area of a rectangle are 25 m and 168 m square respectively. what is the length of the rectangle

Answer 1

Let the length of rectangle is l m and breadth of rectangle is b m,



As all vertex angles of a rectangle are right angles, applying Pythagoras Theorem,


Diagonal of rectangle = √{ l² + b² }


In the question, diagonal of rectangle = 25 m


Comparing both the values of the diagonal,


25 m = √{ l² m² + b² m² }


( 25 m )² = l² m² + b² m²


625m² = l² m² + b² m²


625 m² - b² m² = l² m²


√{ 625 m² - b² m² ) = l m




Area of rectangle = l × b


In the question, area of rectangle = 168 m


Comparing both the values of the area,
.

168 m² = l m × b m


From the above explanation we can write the value of l which is √{ 625 m² - b² m² } ,


185 m² = √{ 625 m² - b² m² } × b m


185 m² = √{ 625 - b² } m × b m


185 m² = √{ 625 - b² } × b m²


185 = √{ 625 - b² } × b


34225 = { 625 - b² } × b


34225 = 625b² - b⁴


b⁴ - 625b² + 34225 = 0



Let b² = x



x² - 625x + 34225 = 0



By applying quadratic formula,


$value \: \: of \: \: x = \dfrac{ 625 \pm 5 \sqrt{10149}}{2}$


So,

$b^{2}= \dfrac{ 625 \pm 5 \sqrt{10149}}{2} \\ \\ b = \sqrt{ \dfrac{ 625 \pm 5 \sqrt{ 10149}}{2}}$



On putting approximate values of roots, we get that value b is 7.78745 or 23.75617




Hence,

b m = Breadth of rectangle = 7.78745 or
23.75617

l m = length of rectangle = √{ 625 m² - ( b )² m² } = √{ 625 m² - ( 7.78745 )² m² } or √{ 625 m² - ( 23.75617 )² } = 23.75617 or 7.78745 respectively and respective to the Breadth.




Answer : length of rectangle = 23.75617 or 7.78745