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A parallelogram has vertices at the coordinates π΄ negative four, negative one; π΅ zero, negative three; πΆ negative one, negative five; and π· negative five, negative three.
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Work out the length of the diagonal π΄πΆ.
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Work out the length of the diagonal π΅π·.
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Using these lengths, is the parallelogram π΄π΅πΆπ· a rectangle?
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So weβve been given the coordinates of the four vertices of a parallelogram and asked to find the length of its two diagonals.
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To do this, weβll need to recall the distance formula, which tells us how to calculate the distance between two points on a coordinate grid.
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If the two points have coordinates π₯ one π¦ one and π₯ two π¦ two, then the distance between them is given by the square root of π₯ two minus π₯ one squared plus π¦ two minus π¦ one squared.
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This is an application of the Pythagorean theorem, where π₯ two minus π₯ one and π¦ two minus π¦ one are the horizontal and vertical sides of a right-angled triangle and π is the hypotenuse.
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To find the length of the diagonal π΄πΆ, we need to substitute the coordinates for π΄ and πΆ into the distance formula.
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Now there are also a lots of negatives involved here, so we need to be careful with the signs.
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We have that π΄πΆ is equal to the square root of negative one minus negative four squared plus negative five minus negative one squared.
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This is equal to the square root of three squared plus negative four squared.
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Three squared is nine and negative four squared is 16.
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So we have the square root of nine plus 16, which is equal to the square root of 25.
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25 is a square number and its square root is exactly equal to five.
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So weβve found the length of the first diagonal π΄πΆ, and now we need to find the length of the second diagonal π΅π·.
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We weβll substitute the coordinates for π΅ and π· into the distance formula.
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Again, we need to be very careful with the negative signs.
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We have that π΅π· is equal to the square root of negative five minus zero squared plus negative three minus negative three squared.
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This simplifies to the square root of negative five squared plus zero squared.
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Negative five squared is 25 and zero squared is zero, so we have the square root of 25 which is equal to five.
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You will have noticed, Iβm sure, that the length of the two diagonals of this parallelogram are the same.
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Theyβre both equal to five units.
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How does this help us with answering the final part of the question?
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Well, a key fact which is true of rectangles but isnβt true of parallelograms in general is that the diagonals are equal in length.
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Weβve already calculated that π΄πΆ and π΅π· are the same length.
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They are both equal to five and, therefore, this tells us that the parallelogram π΄π΅πΆπ· is a rectangle.