find the dimensions of a rectangle having perimeter 14m and diagonal of length 5m
Let the diagonal be c and the sides a and b
b = sqrt (c^2 - a^2) = sqrt(25-a^2)
Perimeter = 2a + 2b = 2a + 2sqrt (25-a^2) = 14
14 - 2a = 2 sqrt (25-a^2)
196 - 56a + 4a^2 = 4(25-a^2) = 100 -4a^2
8a^2 -56a +96 = 0
a^2 -7a + 12 = 0
(a-3)(a-4) = 0
The side lengths are 3 and 4
To find the dimensions of a rectangle with a given perimeter and diagonal, let's break down the problem step by step.
Step 1: Understand the problem.
A rectangle has four sides, and its perimeter is the sum of all four sides. The diagonal of a rectangle is a line segment connecting two nonadjacent corners. We are given that the perimeter of the rectangle is 14m, and the length of the diagonal is 5m. We need to find the dimensions (length and width) of the rectangle.
Step 2: Define the variables.
Let's assume that the length of the rectangle is L (in meters) and the width is W (in meters).
Step 3: Formulate equations.
We know that the perimeter of a rectangle is given by P = 2L + 2W, where P is the perimeter. In this case, P = 14m. So, the equation becomes:
2L + 2W = 14 ---- (Equation 1)
We also know that the length, width, and diagonal of a rectangle form a right-angled triangle, where the diagonal is the hypotenuse. Using the Pythagorean theorem, we can write:
L^2 + W^2 = D^2 ---- (Equation 2)
Substituting D = 5m into Equation 2, we get:
L^2 + W^2 = 5^2
L^2 + W^2 = 25 ---- (Equation 3)
Step 4: Solve the equations.
To solve the equations simultaneously and find the dimensions of the rectangle, we can use the method of substitution.
From Equation 1, we can express L in terms of W:
2L = 14 - 2W
L = 7 - W ---- (Equation 4)
Substituting Equation 4 into Equation 3, we have:
(7 - W)^2 + W^2 = 25
Expanding and simplifying:
49 - 14W + W^2 + W^2 = 25
2W^2 - 14W + 24 = 0
Dividing the equation by 2, we get:
W^2 - 7W + 12 = 0
Factorizing the quadratic equation, we have:
(W - 3)(W - 4) = 0
So, W = 3 or W = 4
If W = 3, substituting this value into Equation 4:
L = 7 - 3 = 4
If W = 4, substituting this value into Equation 4:
L = 7 - 4 = 3
Therefore, the dimensions of the rectangle can be either 3m by 4m or 4m by 3m.