A quadrilateral has vertices at the points two, one; three, three; five, two; and four, zero. By determining the length of the quadrilateral’s sides and considering the gradient of the intersecting lines, what is the name of the quadrilateral?
So we’ve been given the coordinates of the four vertices of a quadrilateral and asked to determine what type of quadrilateral it is. There are of course a number of different possibilities. We’ve been asked to answer the question by determining the length of the quadrilateral sides and then by considering the gradients of the intersecting lines. So we will be thinking about what properties this quadrilateral has in order to determine what type of quadrilateral it is.
Let’s begin by determining the length of the quadrilateral’s four sides. I’ve labelled the four vertices as 𝐴, 𝐵, 𝐶, and 𝐷 so that we can refer to them more easily. In order to calculate the length of each side, we’re going to use the distance formula which tells us that distance between the two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is found by taking the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared.
This is an application of the Pythagorean theorem. So we have four lengths that we need to find: 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴. Let’s begin with the length of 𝐴𝐵. Applying the distance formula, we have that the length of 𝐴𝐵 is the square root of three minus two all squared plus three minus one all squared. This gives the square root of one squared plus two squared. One squared is one and two squared is four. So the length of 𝐴𝐵 is the square root of five.
Next, let’s consider the length of the side 𝐵𝐶. Applying the distance formula for this side gives the square root of five minus three all squared plus two minus three all squared. This gives the square root of two squared plus negative one squared. Two squared is four and negative one squared is one. So overall, this is the square root of five.
So far then, we have two sides of this quadrilateral the same length. Let’s consider the final two sides. The distance formula for the side 𝐶𝐷 gives the square root of four minus five all squared plus zero minus two all squared. This gives the square root of negative one squared plus negative two squared. Negative one squared is one and negative two squared is four. So we have the square root of five again.
Finally, for the side 𝐷𝐴, the distance formula gives the square root of two minus four all squared plus one minus zero all squared. This gives the square root of negative two squared plus one squared which once again gives the square root of five.
So by applying the distance formula, we found that all four sides of the quadrilateral are the same length. They all the square root of five. What does this tell us about what type of quadrilateral we have? Well, of all of the different types of quadrilateral mentioned earlier, there’re only two that have the property that all four sides are the same length. This means the quadrilateral we’re interested in is either a square or a rhombus.
In order to determine which it is, we now need to consider the gradients of the intersecting lines. In a square, all of the vertices are right angles, which means intersecting lines are perpendicular to each other. This means that the product of their slopes which we can refer to as 𝑚 one and 𝑚 two must be equal to negative one. In a rhombus, this isn’t the case.
So by calculation the gradients of the four sides of the quadrilateral, we’ll determine whether or not this relationship exists. To calculate the gradients, we’ll need to apply the slope formula, which tells us that the slope of the line segment joining the points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two can be calculated as the change in 𝑦 over the change in 𝑥. 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.
We have four gradients to calculate. Let’s begin with the side 𝐴𝐵. 𝑦 two minus 𝑦 one is three minus one and 𝑥 two minus 𝑥 one is three minus two. This gives two over one, which simplifies to two. So the slope of the line 𝐴𝐵 is two. Now, let’s consider the next side 𝐵𝐶. For 𝐵𝐶, 𝑦 two minus 𝑦 one is two minus three and 𝑥 two minus 𝑥 one is five minus three. This gives a slope of negative one-half for the line 𝐵𝐶.
So we have the gradients of two of the sides and we need to find the final two. Next, let’s think about the side 𝐶𝐷. The slope is zero minus two over four minus five. This simplifies to negative two over negative one. And the two negatives cancel each other out here, giving a slope of two for the line 𝐶𝐷.
Let’s find the final slope — the slope of 𝐷𝐴. This slope is one minus zero over two minus four. This is equivalent to one over negative two, which is better written as negative one-half. So we have the slopes now of all four of the sides. We can see that opposite sides are parallel. But that’s true in both a square and a rhombus. We need to consider the relationship between the gradients of adjacent sides — the intersecting lines.
All the pairs of intersecting lines have gradients of two and negative a half. If we multiply these two values together, we get negative one, which shows that all pairs of intersecting lines are perpendicular to each other.
So first, we showed that this quadrilateral has four equal side lengths, which means that it could be a rhombus or a square. Next, we calculated the gradients of intersecting lines and showed that all pairs of intersecting lines are perpendicular to one another, which means that this quadrilateral is a square.