Answer :
The ratio of the lengths of the four sides of the trapezoid of 1:1:1:2, and the area of 48·√3, gives the perimeter of the trapezoid as 40
What is a trapezoid in geometry?
A trapezoid is a quadrilateral that has a pair of sides that are parallel.
The given ratio of the sides are: 1:1:1:2
The area of the trapezoid is A = 48·√3
Required: The perimeter of the trapezoid
[tex]Area\ of \ a \ trapezoid \ is, A = \dfrac{a+b}{2} \cdot h[/tex]
Where:
a and b are the lengths of the parallel sides
Given that a ≠ b, when the proportionate length of b = 2, then a = 1
Which gives that the proportionate lengths of the other sides are also 1 and 1
The trapezoid is therefore, an isosceles trapezoid, and the proportionate length of h is therefore;
h² = √(a² - (0.5·a)²) = √(a² - 0.25·a²) = √(0.75·a²)
[tex]h = \dfrac{\sqrt{3} }{2} \cdot a[/tex]
Which gives: [tex]A = \dfrac{a+2\cdot a}{2} \times \dfrac{\sqrt{3} }{2} \cdot a = a\cdot \left(\dfrac{3\cd}{2} + \dfrac{\sqrt{3} }{2} \right) = \dfrac{3\cdot \sqrt{3}\cdot a^2 }{4} =48\cdot \sqrt{3}[/tex]
[tex]\dfrac{3\cdot a^2 }{4} =48[/tex]
3·a² = 48 × 4 = 192
[tex]a^2=\dfrac{192}{3} =64[/tex]
a = √(64) = 8
b = 2·a
∴ b = 2 × 8 = 16
The lengths of sides of the trapezoid are: 8, 8, 8, 16
The perimeter is P = 8 + 8 + 8 + 16 = 40
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