Squares Examples

Is ABCD a square?

ABCD is a rhombus because it is an equilateral quadrilateral. And look! It has a right angle. If we wanted to go through the tedious process of calculating the measure of every angle, we'd see that they're all 90°. It's a square all right.

The quadrilateral ABCD is a square. If AB = x + 1 and  units, what is x?

We know that ΔABC is a right triangle with AB ≅ BC. We just apply the Pythagorean Theorem or our knowledge of special right triangles and solve for x. Shouldn't be too difficult, right?

2(x + 1)² = 18
(x + 1)² = 9
x + 1 = ±3
x = -1 ± 3
x = 2 or x = -4

Because x is a length, it must be positive. So x = 2 is the correct value, which means the sides of the square are 3 units long.

The perimeter of a square is 36 mm long. What is the length of one diagonal?

We know that a square's perimeter is equal to 4 times its side length, or 4s. Solving 4s = 36 for s give us s = 9 mm. Either through the Pythagorean Theorem or our undying knowledge of 45-45-90 triangles, we can find that the diagonal of a square is mm ≈ 12.7 mm long.

Prove that if LOUI is a rhombus and OE ≅ UE, then LOUI is a square.