The perimeter of a rectangle is 24, and its diagonal is √74. Find its dimensions and area.
x+y = 12
x^2+y^2 = 74
(x+y)^2 = x^2+y^2 + 2xy = 144
74+2xy=144
2xy=70
xy=35
35 = 7*5
So, the rectangle is 7x5
To find the dimensions and area of a rectangle given the perimeter and diagonal, you can use the following steps:
Step 1: Recall the formulas for perimeter, diagonal, dimensions, and area of a rectangle:
- Perimeter: P = 2(length + width)
- Diagonal: D = √(length^2 + width^2)
- Area: A = length * width
Step 2: Set up the equations using the given information:
We are given:
- Perimeter, P = 24
- Diagonal, D = √74
We need to find:
- Length, L
- Width, W
- Area, A
Using the perimeter formula, we have:
24 = 2(L + W) (Equation 1)
Using the diagonal formula, we have:
√74 = √(L^2 + W^2) (Equation 2)
Step 3: Solve the system of equations:
From Equation 1, we can rewrite it as:
L + W = 12 (Equation 3)
Squaring Equation 3, we get:
L^2 + 2LW + W^2 = 144 (Equation 4)
From Equation 2, we square both sides:
(√74)^2 = (√(L^2 + W^2))^2
74 = L^2 + W^2 (Equation 5)
Now we have a system of equations:
L^2 + W^2 = 74 (Equation 5)
L^2 + 2LW + W^2 = 144 (Equation 4)
By subtracting Equation 5 from Equation 4, we eliminate the LW term:
(2LW - LW) + (W^2 - W^2) = 144 - 74
L^2 = 70
Take the square root of both sides to find the length:
L = √70
Substitute the value of L into Equation 3 to solve for the width:
√70 + W = 12
W = 12 - √70
Step 4: Calculate the dimensions and area:
Length, L = √70
Width, W = 12 - √70
To find the area, substitute the values of length and width into the area formula:
Area, A = L * W
Calculating the area will give you the final answer.
Note: The decimal approximation of √70 is approximately 8.366. Therefore, the dimensions would be approximately:
Length, L ≈ 8.366
Width, W ≈ 3.634
However, it would be more precise to leave the values in radical form until the final calculation of the area.