Sara used 34 meters of fencing to enclose a rectangular region. To be sure that the region was a rectangle, she measured the diagonals and found that they were 13 meters each. What are the length and width of the rectangle?

Given:

Sara used #34# meters of fencing to enclose a rectangular region.

Hence,

Perimeter of a rectangle as shown below is #34# meters

Hence 2x(Length + Width) = 34 meters

Let us assume that Length = L meters and Width = W meters.

So, #2*(L + W) = 34# meters

What is below is a rough sketch and NOT drawn to scale

Hence,

AB = CD = L meters

AC = BD = W meters

We are given that Diagonals are 13 meters long

We know that,

the diagonals of a rectangle are equal length;

diagonals of a rectangle also bisect each other

What is below is a rough sketch and NOT drawn to scale

Angle #/_ACD# is right-angle

Using Pythagoras Theorem, we can write

#AC^2 + CD^2 = AD^2#

#rArr W^2 + L^2 = 13^2#

#rArr W^2 + L^2 = 169#

Add #-W^2# on both sides to get

#rArr W^2+L^2 - W^2= 169 - W^2#

#rArr cancel (W^2)+L^2 - cancel (W^2)= 169 - W^2#

#rArr L^2 = 169 - W^2#

Taking square root on both sides

#rArr sqrt(L^2) = sqrt(169 - W^2)#

#rArr L = +- sqrt(13^2 -W^2)#

We consider only positive values

#rArr L = sqrt(13^2) -sqrt(W^2)#

#rArr L = 13 -W#

Substitute #color(red) (L = {13 -W})# in #color(blue)({W^2 + L^2} = 169)#

#rArr W^2 + (13-W)^2 = 169#

Using the identity #color(green)((a-b)^2 -= a^2 - 2ab + b^2)# we get

#rArr W^2 + 169 - 26W + W^2 = 169#

#rArr W^2 + cancel 169 - 26W + W^2 = cancel 169#

#rArr 2W^2 - 26W =0#

#rArr 2W(W -1 3)=0#

#rArr W-13=0#

Hence, #W = 13#

Hence, width of the rectangle = #13# meters

We already have

#2*(L + W) = 34# meters

Substitute the value of #W = 13# to get

#2*(L + 13) = 34#

#rArr 2L + 26 = 34#

Add #-26# to both sides

#rArr 2L + cancel 26 - cancel 26 = 34 - 26#

#rArr 2L = 34 - 26= 8#

#rArr 2L = 8#

#L = 8/2 = 4#

Length of the rectangle = 4 meters

Substitute the values of #L = 4 and W = 13# in

#2*(L + W) = 34# meters

to verify our results

We get

#2*(4 + 13) = 34# meters

#rArr 34 = 34#

Hence, our rectangle has

Length(L) #= 4# meters

Width(W) #= 13# meters