Hi, and welcome to this video on diagonals!
Today we’re going to explore the diagonals of parallelograms, rectangles, rhombi, and squares,
and see how these shapes relate to each other overall.
Let’s get started!
All of the shapes that we are going to look at today are quadrilaterals, meaning they
are all four-sided polygons.
A polygon is a shape with multiple sides.
A parallelogram is a quadrilateral that has two sets of parallel sides.
The blue arrows denote which sides are parallel to each other.
The opposite sides of a parallelogram, marked by green tick marks, are congruent,
which means they have the same measure.
A rectangle is a shape we all know well since it appears in the real world so often.
But one thing we don’t usually think about is that rectangles are just a special kind of parallelogram.
That means that the opposite sides are both parallel and congruent, just like our first parallelogram.
What makes a rectangle a special parallelogram is that its interior angles are all the same.
In fact, they are all right angles, which means they measure 90 degrees.
Another special parallelogram is a rhombus.
It has all the properties of the parallelogram but all of the sides of a rhombus are congruent.
Note that the tick marks are all the same, which tells us that all the sides are the same length.
Our last special parallelogram is a rectangle and a rhombus at the same time, so it has
the right angles of the rectangle and the congruent sides of the rhombus.
It’s another shape we all know well...the square!
As you can see, the opposite sides are parallel, all the angles are right angles, and all the sides are congruent.
So let’s recap quadrilaterals before we start exploring their diagonals.
Here, we’ve arranged our quadrilaterals into a sort of “tree”, starting with the square as the trunk.
Every square is a rhombus and every rhombus is a parallelogram.
Every square is also a rectangle and every rectangle is a parallelogram.
That’s how it works going up the tree.
But it doesn’t work going down.
Not every parallelogram is a rectangle and not every rhombus is a square.
Okay, let’s explore the diagonals of these four quadrilaterals.
A diagonal is a line segment connecting opposite vertices, or corners, of a quadrilateral.
Here’s our regular, non-special parallelogram with the diagonals shown.
As you can see, the diagonals of a parallelogram bisect each other.
In other words, they cut each other in half.
We could add some numbers to illustrate this better,
or we could use tick marks to show that the diagonals have been bisected.
Since the three special quadrilaterals we’re talking about are all parallelograms,
they will have bisecting diagonals.
Now let’s look at the rectangle.
Here’s a rectangle with its diagonals.
The diagonals of a rectangle are congruent, and, again, since a rectangle is a parallelogram,
the diagonals bisect each other, making each half the same length.
Each diagonal of a rectangle also divides the rectangle into two congruent right triangles.
This means that, if we wanted to, we could calculate the length of the diagonal using the Pythagorean Theorem.
Now that we’ve got rectangles figured out, let’s see what happens with another special parallelogram,
the rhombus.
Once again, since every rhombus is a parallelogram the diagonals bisect each other.
The diagonals are not the same size though, so what’s special about this one?
Take a look at the angles at which the diagonals intersect.
These angles look like they could all be the same, and since there are four angles there it must mean
that each angle is 90 degrees!
This means that the diagonals of a rhombus are perpendicular to each other in addition
to bisecting each other.
Okay, only one quadrilateral left, the square.
Remember, the square is a parallelogram, a rectangle, and a rhombus, so it should have
all the properties of those shapes.
1. The diagonals will bisect each other.
2. The diagonals will be the same length.
and 3. The diagonals will be perpendicular to each other.
Let’s see if we’re right.
We were right!
If we look closely, we can also see that the two diagonals cut the square into four congruent
isosceles right triangles.
Now that we’ve looked at our four parallelograms and their diagonals, let’s finish with a review
to see what all you can remember.
Which of the following statements is false?
A. Every rectangle is a parallelogram.
B. Every rhombus is a square.
C. Every square is a rectangle.
D. The diagonals of a parallelogram bisect each other.
E. The diagonals of a rhombus intersect at right angles.
F. A diagonal of a rectangle divides it into two congruent right triangles.
G. The diagonals of a rectangle are the same length.
H. A quadrilateral whose diagonals bisect each other, intersect at right angles, and are
congruent must be a square.
The correct answer is B.
Remember the tree?
Not every rhombus is a square, but every square is a rhombus.
Thanks for watching, and happy studying!