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The points πΎ negative five, zero; πΏ negative three, negative one; π negative two, five; and π negative four, six are the vertices of quadrilateral πΎπΏππ.
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Using the slope formula, is the quadrilateral a parallelogram?
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So weβve been given the coordinates of the four vertices of a quadrilateral and asked to determine whether or not this quadrilateral is a parallelogram.
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Weβre also told how to do this using the slope formula.
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Letβs recall its definition.
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The slope of the line joining the points with coordinates π₯ one, π¦ one and π₯ two π¦, two can be found by calculating the change in π¦ divided by the change in π₯.
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π¦ two minus π¦ one over π₯ two minus π₯ one.
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How would this help us with answering the question?
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Well, in order for a quadrilateral to be a parallelogram, it needs to have two pairs of parallel sides β each of its pairs of opposite sides.
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In the case of a parallelogram drawn on a coordinate grid as we have here, this means that the slopes of opposite sides need to be the same.
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So what we need to do is calculate the slope of each side.
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Letβs begin first of all with the side πΎπΏ.
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π¦ two minus π¦ one is negative one minus zero and π₯ two minus π₯ one is negative three minus negative five.
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This gives the slope of πΎπΏ as negative a half.
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Next, letβs find the slope of the side πΏπ.
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The change in π¦ is five minus negative one and the change in π₯ is negative two minus negative three.
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This simplifies to six over one, which is just six.
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Now, these are a pair of adjacent sides.
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So weβre not expecting them to have the same slope.
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If indeed, this quadrilateral is a parallelogram.
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Letβs consider the final two sides.
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Firstly, side ππ, the slope is six minus five over negative four minus negative two.
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This simplifies to one over negative two, which is better written as negative a half.
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Now, notice this is the same as the slope weβve already calculated for the side πΎπΏ.
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So it is true that the lines πΎπΏ and ππ are parallel.
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Therefore, our quadrilateral does have at least one pair of parallel sides.
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Letβs consider the slope of the final side ππΎ.
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The slope is zero minus six over negative five minus negative four.
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This simplifies to negative six over negative one, which is just equal to six.
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Again, notice that this is the same as the slope of already calculated for the side πΏπ.
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So now we know that πΏπ is parallel to ππΎ.
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So weβve shown that the quadrilateral πΎπΏππ has two pairs of parallel sides.
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And therefore, our answer to the question βis it a parallelogram?β is yes.