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