# A rectangular parcel of land is 50 feet longer than it is wide. Each diagonal between opposite...

## Question:

A rectangular parcel of land is 50 feet longer than it is wide. Each diagonal between opposite corners is 250 feet.

What are the dimensions of the parcel?

## Diagonals and Sides of a Rectangle:

Rectangle - It is a planner shape that has four sides and four interior angles such that opposite sides are parallel and equal to reach other and each angle is a right angle.

It has four vertices or corners.

The distance between the opposite vertices or corners is called the diagonal length of the rectangle.

A rectangle has two diagonals having the same length.

#### Diagonal Length

It is calculated using the Pythagorean theorem-

{eq}\displaystyle D^{2} = L^{2}+W^{2} {/eq}

here

'L' is the length

'W' is the width

'D' is the length of the diagonal of the rectangle.

#### Area

{eq}\displaystyle A = L \times W {/eq}

here

A is the area of the rectangle

#### Perimeter

{eq}\displaystyle P = 2(L+W) {/eq}

here P is the perimeter of the rectangle

Given that a rectangular parcel of land is 50 feet longer than it is wide.

{eq}\displaystyle L = 50+W -------(1) {/eq}

Also given that the diagonal length of the parcel is 250 feet.

{eq}\displaystyle D = 250 ~ft {/eq}

Using the Pythagorean theorem-

{eq}\displaystyle 250^{2} = L^{2}+W^{2} {/eq}

now put the value of L from the equation(1)-

{eq}\displaystyle 62500 = (50+W)^{2}+W^{2} {/eq}

{eq}\displaystyle 62500 = 2500+100W+W^{2}+W^{2} {/eq}

{eq}\displaystyle 2W^{2}+100W-60000= 0 {/eq}

{eq}\displaystyle W^{2}+50W-30000 =0 {/eq}

by factorig

{eq}\displaystyle W^{2}+200W-150W -30000= 0 {/eq}

{eq}\displaystyle W(W+200)-150(W+200) = 0 {/eq}

{eq}\displaystyle (W+200)(W-150) = 0 {/eq}

So

{eq}\displaystyle W = 150 {/eq}

or

{eq}\displaystyle W = -200 {/eq}

because W is the width of the rectangle it can not be negative.

So

{eq}\displaystyle \Rightarrow W = 150 ~ft {/eq}

from the equation(1)-

{eq}\displaystyle L = 50+150 = 200 ~ft {/eq}

So the length and width of the given rectangle are {eq}200 ~ft {/eq} and {eq}150~ ft {/eq} respectively.