The perimeter of an isosceles trapezium is 134 cm and the bases are 54 cm and 30 cm in length.find the length of nonparallel sides of the trapezium and its area.
the unknown sides are equal, say x. So,
54+x+30+x = 134
If you draw your trapezium and mark off the interior 30x30 square, you will find a right triangle at each end, with base 12 and hypotenuse x.
So, the height of the trapezium is √(x^2-144) making the area
A = (30+54)/2 √(x^2-144)
To find the length of the nonparallel sides of the trapezium, we can start by assigning variables to the lengths we need to find.
Let's say that the length of the nonparallel side is 'x'. By definition, an isosceles trapezium has two nonparallel sides of equal length. So, the other nonparallel side will also be 'x'.
The perimeter of a trapezium is calculated by adding all the sides together. In this case, we have:
Perimeter = Length of base 1 + Length of base 2 + Length of nonparallel side 1 + Length of nonparallel side 2
Given:
Perimeter = 134 cm
Length of base 1 = 54 cm
Length of base 2 = 30 cm
So, we can write the equation:
134 = 54 + 30 + x + x
Combine like terms:
134 = 84 + 2x
Subtract 84 from both sides:
50 = 2x
Divide by 2:
x = 25
Therefore, the length of the nonparallel sides of the trapezium is 25 cm each.
To find the area of the trapezium, we can use the formula:
Area = (Sum of the lengths of the bases / 2) × Height
Given:
Length of base 1 = 54 cm
Length of base 2 = 30 cm
To find the height, we can use the following relation:
Height^2 = (Length of base 1 - Length of base 2)^2 - (Length of nonparallel side)^2
Substituting the given values:
Height^2 = (54 - 30)^2 - (25)^2
Height^2 = 24^2 - 25^2
Height^2 = 576 - 625
Height^2 = -49
Since the result is negative, it means there is no valid height for this trapezium. Therefore, we cannot calculate the area.
In summary:
Length of nonparallel sides: 25 cm each
Area: Not calculable due to the absence of a valid height.