Polygons and PolyhedraQuadrilaterals
In the previous course we investigated many different properties of triangles. Now let’s have a look at quadrilaterals.
A regular quadrilateral is called a
For slightly “less regular” quadrilaterals, we have two options. If we just want the angles to be equal, we get a
There are a few other quadrilaterals, that are even less regular but still have certain important properties:
Quadrilaterals can fall into multiple of these categories. We can visualise the hierarchy of different types of quadrilaterals as a
For example, every rectangle is also a
To avoid any ambiguity, we usually use just the most specific type.
Now pick four points, anywhere in the grey box on the left. We can connect all of them to form a quadrilateral.
Let’s find the midpoint of each of the four sides. If we connect the midpoints, we get
Try moving the vertices of the outer quadrilateral and observe what happens to the smaller one. It looks like it is not just any quadrilateral, but always a
But why is that the case? Why should the result for any quadrilateral always end up being a parallelogram? To help us explain, we need to draw one of the
The diagonal splits the quadrilateral into two triangles. And now you can see that two of the sides of the inner quadrilateral are actually
In the previous course we showed that
We can do exactly the same with the second diagonal of the quadrilateral, to show that both pairs of opposite sides are parallel. And this is all we need to prove that the inner quadrilateral is a
It turns out that parallelograms have many other interesting properties, other than opposite sides being parallel. Which of the following six statements are true?
Of course, simply “observing” these properties is not enough. To be sure that they are always true, we need to prove them:
Opposite Sides and Angles
Let’s try to prove that the opposite sides and angles in a parallelogram are always congruent.
Start by drawing one of the diagonals of the parallelogram.
The diagonal creates four new angles with the sides of the of the parallelogram. The two red angles and the two blue angles are
Now if we look at the two triangles created by the diagonal, we see that they have two congruent angles, and one congruent side. By the
This means that the other corresponding parts of the triangles must also be congruent: in particular, both pairs of opposite sides are congruent, and both pairs of opposite angles are congruent.
It turns out that the converse is also true: if both pairs of opposite sides (or angles) in a quadrilateral are congruent, then the quadrilateral has to be a parallelogram.
Now prove that the two diagonals in a parallelogram bisect each other.
Let’s think about the two yellow triangles generated by the diagonals:
- We have just proved that the two green sides are congruent, because they are opposite sides of a parallelogram.
- The two red angles and two blue angles are congruent, because they are
alternate anglesopposite anglesright angles.
Now we can use the fact the corresponding parts of congruent triangles are also congruent, to conclude that
Like before, the opposite is also true: if the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
We showed above that the two pairs of
The name Kite clearly comes from its shape: it looks like the kites you can fly in the sky. However, of all the special quadrilaterals we have seen so far, the Kite is the only one that can also be
You might have noticed that all kites are
The diagonal splits the kite into two congruent triangles. We know that they are congruent from the
This means, for example, that the diagonal is a
We can go even further: if we draw the other diagonal, we get two more, smaller triangles. These must also be congruent, because of the
This means that angle α must also be the same as angle β. Since they are adjacent,
In other words, the diagonals of a kite are always
Area of Quadrilaterals
When calculating the area of triangles in the previous course, we used the trick of converting it into a
On the left, try to draw a rectangle that has the same area as the parallelogram.
Can you see that the missing triangle on the left is
Area = base × height
Be careful when measuring the height of a parallelogram: it is usually not the same as one of the two sides.
Recall that trapeziums are quadrilaterals with one pair of parallel sides. These parallel sides are called the bases of the trapezium.
Like before, try to draw a rectangle that has the same area as this trapezium. Can you see how the missing and added triangles on the left and the right cancel out?
The height of this rectangle is the
The width of the rectangle is the distance between the
If we combine all of this, we get an equation for the area of a trapezium with parallel sides a and c, and height h:
In this kite, the two diagonals form the width and the height of a large rectangle that surrounds the kite.
The area of this rectangle is
This means that the area of a kite with diagonals d1 and d2 is
This means that to find the area of a rhombus, we can use either the equation for the area of a parallelogram, or that for the area of a kite:
Area = base × height =
In different contexts, you might be given different parts of a Rhombus (sides, height, diagonals), and you should pick whichever equation is more convenient.