Question

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?

Answer 1

Length(L) `= 4` meters

Width(W) `= 13` meters

Explanation:

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