Question

Guys please answer me soon with an easy explanation

the diagonal of a rectangle is 20 metre and its parameter is 50 metre then what are its dimensions?

Answer 1

"Parameter" = Perimeter.

Look at the picture.

We have the perimeter = 50 m.

The perimeter is 2l + 2w (l - length, w - width). Therefore

2l + 2w = 50 divide both sides by 2

l + w = 25 subtract w from both sides

l = 25 - w.

Use the Pythagorean theorem:

`l^2+w^2=20^2\to(25-w)^2+w^2=20^2`

Use (a - b)² = a² - 2ab + b²

`25^2-2(25)(w)+w^2+w^2=400\\\\625-50w+2w^2=400\qquad\text{subtract 400 from both sides}\\\\225-50w+2w^2=0\\\\2w^2-50w+225=0`

Use quadratic formula:

`ax^2+bx+c=0\\\\\Delta=b^2-4ac\\\\x_1=(-b-\sqrt\Delta)/(2a);\ x_2=(-b+\sqrt\Delta)/(2a)`

We have:

`a=2,\ b=-50,\ c=225`

Substitute:

`\Delta=(-50)^2-4(2)(225)=2500-1000=1500\\\\\sqrt\Delta=√(1500)=√(100\cdot15)=√(100)\cdot√(15)=10√(15)\\\\w_1=(-(-50)-10√(15))/(2(2))=(50-10√(15))/(4)=(25-5√(15))/(2)\\\\w_2=(-(-50)+10√(15))/(2(2))=(50+10√(15))/(4)=(25+5√(15))/(2)`

`l_1=25-w_1\\\\l_1=25-(25-5√(15))/(2)=(50)/(2)-(25-5√(15))/(2)=(50-25+5√(15))/(2)=(25+5√(15))/(2)\\\\l_2=25-w_2\\\\l_2=25-(25+5√(15))/(2)=(50)/(2)-(25+5√(15))/(2)=(50-25-5√(15))/(2)=(25-5√(15))/(2)`

`Answer:\ \boxed{(25+5√(15))/(2)\ m*(25-5√(15))/(2)\ m}`

Answer 2

Length 19.11 and width 5.89.

Explanation:

Let the length be x and width be y metres.

Then, using the Pythagoras theorem:-

x^2 + y^2 = 20^2 = 400....................(1)

The perimeter = 50 so:-

2x + 2y = 50

Dividing through by 2:-

x + y = 25 .............................(2)

So y = 25 - x

Substitute for y in equation (1):-

x^2 + (25 - x)^2 = 400

x^2 + 625 - 50x + x^2 = 400

2x^2 - 50x + 225 = 0

x = 19.11 , 5.89, x = 19.11 as its the length

and y = 25 - 19.11 = 5.89 ( from equation (2).