A quadrilateral has vertices at the point two, one; three, three; six, one; and five, negative one. What is the name of the quadrilateral?
To start with, let’s draw a coordinate grid and plot these points. So here we have our coordinates plotted and lines joining them. Let’s have a look at the properties. If we have a look at the line joining the coordinates three, three and six, one, it goes horizontally three units and vertically two units. The line between the coordinate two, one and five, negative one also goes three units horizontally and two units vertically. So we have a set of parallel lines.
Next, if we look at the line between the coordinate six, one and five, negative one, this has a vertical component of two units and a horizontal component of one unit, which is the same as the line joining the coordinates three, three and two, one. So we have another pair of parallel lines. Therefore, we could say that our quadrilateral has two pairs of parallel lines.
Quadrilaterals that have this property are a square, a rectangle, a rhombus, and a parallelogram. However, in the case of a square and a rhombus, these are quadrilaterals that have all four sides the same length. We have seen in our quadrilateral that this is not the case. Eliminating those two then, we can see that we’re left with the choices of a rectangle or a parallelogram. The difference between a rectangle and a parallelogram is that a rectangle has 90-degree angles. So let’s see what we can say about the angles in our quadrilateral.
We can check for right angles using the Pythagorean theorem. This tells us that the square of the hypotenuse is equal to the sum of the squares of the other two sides. So if we have the hypotenuse to find as 𝑐 and the other two sides as 𝑎 and 𝑏, then 𝑐 squared equals 𝑎 squared plus 𝑏 squared. So let’s create a triangle within our quadrilateral and check if the angle at the coordinate three, three is a right angle. To find our length 𝑎 then, we can see that that’s created by going three horizontally and two vertically. So our hypotenuse 𝑎 squared is equal to three squared plus two squared. So 𝑎 squared equals nine plus four, which is 13. So 𝑎 is equal to root 13.
In the same way, to find our side length 𝑏, using the Pythagorean theorem, we can say that 𝑏 squared is equal to one squared plus two squared, which is one plus four. Therefore, 𝑏 squared equals five. So 𝑏 is equal to root five. So to see if the Pythagorean theorem applies in our orange triangle, we need the final missing horizontal length. In this case, it’s a simple horizontal line. And we can see that it would be four units long as the 𝑥-value in our coordinates goes from two to six.
So now, in our orange triangle, we check if the Pythagorean theorem works. Do we have four squared equal to root 13 squared plus root five squared? Well, if we square the square root of a number, then we end up with just the number. So root 13 squared is 13. In the same way, the square of root five would be five. We can evaluate our left-hand side, four squared, as 16. So does 16 equal 13 plus five? No, because 16 is not equal to 18. This means that the Pythagorean theorem doesn’t work. So our shape has no right angles and is therefore not a rectangle.
We can then conclude that the quadrilateral given by these coordinates is a parallelogram.