The sides of a pallelogram in fig 14.6 are given in cm with side is 2x+y, 4x-3, 3y+2x, 5y-8. Find x and y and the perimeter of the parallelogram
To find x and y, we need to equate the opposite sides of the parallelogram:
2x + y = 3y + 2x
4x - 3 = 5y - 8
Simplifying the first equation, we get:
2x - 2x = 3y - y
0 = 2y
Therefore, y = 0.
Using this value of y in the second equation:
4x - 3 = -8
4x = -8 + 3
4x = -5
x = -5/4
So, x = -5/4 and y = 0.
Now, let's find the perimeter of the parallelogram.
The length of one side of the parallelogram is 2x + y = 2(-5/4) + 0 = -5/2.
The length of the adjacent side is 4x - 3 = 4(-5/4) - 3 = -5 - 3 = -8.
The other two sides can be found using the same approach.
So, the perimeter of the parallelogram is:
Perimeter = (2)(-5/2) + (-8) + (3)(0) + (5)(0)
Perimeter = -5 - 8 + 0 + 0
Perimeter = -13 + 0 + 0
Perimeter = -13.
Therefore, the perimeter of the parallelogram is -13 cm.
Note: Typically, lengths and perimeters are expressed as positive values.
To find the values of x and y, we can equate the given expressions for the sides of the parallelogram.
The sides of the parallelogram are:
Side 1: 2x + y
Side 2: 4x - 3
Side 3: 3y + 2x
Side 4: 5y - 8
Since opposite sides of a parallelogram are equal in length, we can equate the expressions for the sides:
2x + y = 4x - 3 (Equation 1)
3y + 2x = 5y - 8 (Equation 2)
Let's solve Equation 1 for x:
2x + y = 4x - 3
Bringing all the x terms to one side and the constant terms to the other side:
2x - 4x = - 3 - y
-2x = -3 - y
2x = y + 3 (Equation 3)
Now, let's solve Equation 2 for y:
3y + 2x = 5y - 8
Bringing all the y terms to one side and the constant terms to the other side:
3y - 5y = -8 - 2x
-2y = -8 - 2x
2y = 8 + 2x (Equation 4)
We now have two equations, Equation 3 and Equation 4, which describe the relationship between x and y. We can now solve these equations simultaneously to find the values of x and y.
Let's substitute the value of y from Equation 3 into Equation 4:
2y = 8 + 2x
2(y + 3) = 8 + 2x
2y + 6 = 8 + 2x
2y = 2x + 2
Now, substitute this value of 2y into Equation 4:
2x + 2 = 8 + 2x
2 = 8
This equation is not true for any value of x or y, so there is no solution.
Therefore, we cannot find the values of x and y.
As for the perimeter of the parallelogram, we cannot determine it without the values of x and y.
To find the values of x and y, we need to set up a system of equations using the given information about the sides of the parallelogram:
Side 1: 2x + y
Side 2: 4x - 3
Side 3: 3y + 2x
Side 4: 5y - 8
Since our parallelogram has opposite sides that are equal in length, we can set up the following equations:
2x + y = 4x - 3 [Equation 1]
3y + 2x = 5y - 8 [Equation 2]
We can solve this system of equations to find the values of x and y. Let's solve it step by step:
Step 1: Simplify Equation 1
2x + y = 4x - 3
Subtract 2x from both sides:
y = 2x - 3x - 3
y = -x - 3 [Equation 3]
Step 2: Simplify Equation 2
3y + 2x = 5y - 8
Subtract 3y from both sides:
2x = 2y - 8
Divide both sides by 2:
x = y - 4 [Equation 4]
Now we have two equations:
y = -x - 3 [Equation 3]
x = y - 4 [Equation 4]
We can substitute Equation 4 into Equation 3 to solve for y:
y = -(y - 4) - 3
y = -y + 4 - 3
2y = 1
y = 1/2
Now we substitute the value of y back into Equation 4 to solve for x:
x = (1/2) - 4
x = -7/2
So, x = -7/2 and y = 1/2.
To find the perimeter of the parallelogram, we sum up the lengths of all four sides:
Perimeter = Side 1 + Side 2 + Side 3 + Side 4
Perimeter = (2x + y) + (4x - 3) + (3y + 2x) + (5y - 8)
Now, substitute the values we found for x and y:
Perimeter = (2*(-7/2) + 1/2) + (4*(-7/2) - 3) + (3*(1/2) + 2*(-7/2)) + (5*(1/2) - 8)
Simplify the expression:
Perimeter = (-7 + 1/2) + (-14 - 3) + (3/2 - 7) + (5/2 - 8)
Perimeter = (-14/2 + 1/2) + (-17) + (-11/2) + (-16/2)
Perimeter = (-13/2) + (-17) + (-11/2) + (-8)
Perimeter = -13/2 - 34/2 - 11/2 - 16/2
Perimeter = -74/2
Perimeter = -37
Since a negative perimeter doesn't make sense, please double-check the given side lengths of the parallelogram or re-verify the equations to get the correct solution.