A rectangle has area 6 cm^2 and diagonal length 2 square root 5cm. What is its perimeter
Let the length and width be x and y, both x and y positive
xy = 6 and x^2 + y^2 = (2√5)^2
from xy=6, y = 6/x
x^2 + (6/x)^2 = 20
x^2 + 36/x^2 = 20
x^4 + 36 = 20x^2
x^4 - 20x^2 + 36 = 0
(x^2 - 18)(x^2 - 2) = 0
x^2 = 18
x = √18 = 3√2, and y = 6/√18 = √2
similarly if x^2 = 2 , then x = √2 and y = 3√2
the perimeter = 2x + 2y
= 6√2 + 2√2 = 8√2
check:
area = (3√2)(√2) = 3(2) = 6 , as given
diagonal:
d^2 = x^2 + y^2
= 18 + 2
= 20
= 2√5 , as given
Find x if the area of the figure shown is A=2 square root of 6cm square
To find the perimeter of a rectangle, we need to know its dimensions. In this case, we are given the area and the diagonal length of the rectangle.
Let's start by finding the length and width of the rectangle.
We know that the formula for the area of a rectangle is given by:
Area = length × width
Given that the area is 6 cm^2, we can express this as:
6 = length × width
Next, we can use the diagonal length to find the relationship between the length and the width.
In a rectangle, the diagonal bisects it into two congruent right triangles. We can use the Pythagorean theorem to relate the diagonal length (hypotenuse) with the length and width:
(diagonal length)^2 = (length)^2 + (width)^2
Substituting in the values we have:
(2√5)^2 = length^2 + width^2
4 × 5 = length^2 + width^2
20 = length^2 + width^2
Now, we have a system of two equations with two variables:
1) 6 = length × width
2) 20 = length^2 + width^2
We can solve this system of equations to find the values of length and width.
From equation 1), we can express length in terms of width:
length = 6 / width
Substituting length into equation 2), we get:
20 = (6 / width)^2 + width^2
20 = 36 / width^2 + width^2
20 = 36 + width^4 / width^2
20 = 36 + 1 / width^2
20 = 36 + 1 / width^2
20 = 36(1 + 1 / width^2)
Now, let's solve for width:
20(1 + 1 / width^2) = 36
1 + 1 / width^2 = 36 / 20
1 + 1 / width^2 = 9 / 5
1 / width^2 = 9 / 5 - 1
1 / width^2 = 4 / 5
width^2 = 5 / 4
width = √(5 / 4)
Since we know that the width cannot be negative, we can take the positive square root:
width = √5 / 2
Now, substituting the value of the width back into equation 1), we can find the length:
length = 6 / width
length = 6 / (√5 / 2)
length = 12 / √5
length = (12√5) / 5
Now that we have found the length and width of the rectangle, we can find its perimeter using the formula:
Perimeter = 2(length + width)
Perimeter = 2[((12√5) / 5) + (√5 / 2)]
Perimeter = (24√5 + 5√5) / 10
Perimeter = (29√5) / 10
Therefore, the perimeter of the rectangle is (29√5) / 10 cm.