Question 239058
P = 2L + 2W = 82
D = 29


Diagonal forms right triangle with sides of the rectangle.


This means that:


L^2 + w^2 = D^2 which means that:


L^2 + W^2 = 29^2


Since 2L + 2W = 82, this means that:


2L = 82-2W which means that:


L = 41 - W


L^2 + W^2 = 29^2 becomes:


(41-W)^2 + W^2 = 29^2 which becomes:


41^2 - 82W + W^2 + W^2 = 29^2 which becomes:


41^2 - 82W + W^2 + W^2 - 29^2 = 0


combine like terms and simplify to get:


2W^2 - 82W + 840 = 0


divide both side by 2 to get:


W^2 - 41W + 420 = 0


This factors out to:


(W-20) * (W - 21) = 0


This makes W = 20 or W = 21


If W = 20, then L = 21


If W = 21, then L = 20


Your answers are:


L = 20 and W = 21
or:
L = 21 and L = 20


To confirm, plug these values into the original equations and see if the equations are true.


Assume L = 20 and W = 21.


L^2 + W^2 = 29^2 becomes 20^2 + 21^2 = 29^2 which becomes 841 = 841 which is true.


Making L = 21 and W = 20 will yield the same answer.


2L + 2W = 82 becomes:


40 + 42 = 82 which is also true.


The values shown look good.


Answers are:


L = 20 and W = 21
or:
L = 21 and W = 20