Rectangles Examples

If a quadrilateral is a rectangle, is it also a parallelogram? Why?

Yes. The definition of a rectangle is that it is a parallelogram with at least one right angle. A rectangle is a special type of parallelogram.

Given that HANK is a rectangle, prove that his diagonals, AK and HN are congruent to each other.

Let's start by drawing a rectangle with its diagonals.

Is the quadrilateral in the figure a rectangle? Why?

Well, it's pretty obvious that it isn't, but what's our reasoning? We can define a rectangle as a parallelogram with at least one right angle. This quadrilateral has one right angle for sure, but it's definitely not a parallelogram.

ABCD is a rectangle. What is x?

Since ABCD is a rectangle, all four internal angles are right angles, including the dude up there in the upper-left corner.

Because it's also a parallelogram, we know that the diagonals bisect each other. So the length of the hypotenuse of one of the right triangles is 2x. To find x, we just have to find that hypotenuse.

Using the Pythagorean Theorem, we get:

a² + b² = c²
5² + 10² = (2x)²
125 = 4x²
31.25 = x²
x ≈ 5.59 units