abcd is a rectangle such that ab=-3 radical sign 48+2 radical sign 75+4radical sign108 and Bc=4 radical sign 27 +radical sign 300 prove that abcd is a square

To prove that ABCD is a square, we need to show that all four sides of the rectangle are equal in length.

Given that AB = -3√48 + 2√75 + 4√108 and BC = 4√27 + √300, we can simplify these expressions to find their values.

Let's simplify AB first:
AB = -3√48 + 2√75 + 4√108

First, simplify the square roots:
√48 = √(16 * 3) = 4√3
√75 = √(25 * 3) = 5√3
√108 = √(36 * 3) = 6√3

Now substitute these values back into the equation:
AB = -3(4√3) + 2(5√3) + 4(6√3)
AB = -12√3 + 10√3 + 24√3
AB = 22√3

Now, let's simplify BC:
BC = 4√27 + √300

First, simplify the square roots:
√27 = √(9 * 3) = 3√3
√300 = √(100 * 3) = 10√3

Now substitute these values back into the equation:
BC = 4(3√3) + 10√3
BC = 12√3 + 10√3
BC = 22√3

Now that we have the values of AB and BC, we can compare them:
AB = 22√3
BC = 22√3

As both AB and BC have the same value, we can conclude that AB = BC. Since the opposite sides of a rectangle are equal, we can also conclude that AD = CD, making ABCD a square.

Therefore, we have proven that ABCD is a square.