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.