The point of intersection of the diagonals of a rectangle is 4 cm further away from the smaller side than from the larger side of the rectangle. The perimeter of the rectangle is equal to 56 cm. Find the lengths of the sides of the rectangle.
Since the diagonals intersect halfway between the sides, we have the
width = 2x
length = 2(x+4)
Now use those to get the perimeter:
2(2x+2(x+4))=56
Solve for x, and then you are home free.
x=5
Let's assume that the length of the rectangle is L cm and the width is W cm.
The perimeter of a rectangle is given by the formula P = 2L + 2W.
Given that the perimeter of the rectangle is 56 cm, we can write the following equation:
2L + 2W = 56 (Equation 1)
The diagonals of a rectangle bisect each other and are equal in length. Let's denote the point of intersection of the diagonals as O.
Given that the point O is 4 cm further away from the smaller side than from the larger side, we can form the following equation using the Pythagorean theorem:
(L/2)^2 + (W/2)^2 = [(L+4)/2]^2 + [(W-4)/2]^2
Simplifying this equation, we get:
L^2 + W^2 = (L+4)^2 + (W-4)^2 (Equation 2)
Expanding the terms in Equation 2, we have:
L^2 + W^2 = L^2 + 8L + 16 + W^2 - 8W + 16
Cancelling out the common terms, we get:
8L - 8W = 32 (Equation 3)
Now we have a system of equations consisting of Equation 1 and Equation 3. We can solve this system to find the values of L and W.
Let's solve the system using the substitution method:
From Equation 3, we can express L in terms of W:
L = W + 4
Substitute this value of L into Equation 1:
2(W+4) + 2W = 56
Simplifying this equation, we get:
4W + 8 + 2W = 56
Combining like terms, we get:
6W + 8 = 56
Subtracting 8 from both sides, we get:
6W = 48
Dividing both sides by 6, we get:
W = 8
Substituting this value of W into L = W + 4, we get:
L = 8 + 4 = 12
Therefore, the length of the rectangle is 12 cm and the width is 8 cm.
To solve this problem, we can set up a system of equations based on the information given.
Let's assume the length of the rectangle is L and the width is W.
First, let's find the equation representing the given information about the intersection point of the diagonals. The intersection point divides each diagonal into two equal segments. Since the diagonals of a rectangle are equal in length, the two segments from the intersection point to the longer side will be equal, as will the two segments to the shorter side.
According to the problem, the point of intersection is 4 cm further away from the smaller side than from the larger side. So, one segment from the intersection point to the shorter side will be W/2 + 4 cm, and one segment to the longer side will be L/2 - 4 cm.
Now, let's find the equation representing the perimeter of the rectangle. The perimeter of a rectangle is equal to 2 times the length plus 2 times the width:
2L + 2W = 56
We now have two equations:
L/2 - 4 = W/2 + 4 (1)
2L + 2W = 56 (2)
We can solve this system of equations to find the values of L and W.
Let's simplify equation (1) by multiplying everything by 2:
L - 8 = W + 8
Now, let's rewrite equation (2) in terms of L:
2L = 56 - 2W
Substituting the value of 2L from equation (2) into equation (1):
56 - 2W = 8 + W + 8
Now, simplify and solve for W:
48 = 3W
W = 16 cm
Substituting the value of W back into equation (2) to find L:
2L + 2(16) = 56
2L + 32 = 56
2L = 24
L = 12 cm
Therefore, the lengths of the sides of the rectangle are 12 cm and 16 cm, respectively.