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Given a rhombus whose diagonalsโ lengths are 5.3 centimeters and 3.8 centimeters and a square whose diagonal length is 5.3 centimeters, which one is smaller in area?
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So at first, we might think, okay, we need to know how to find the area of a rhombus.
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That area is ๐ times ๐ divided by two, where ๐ and ๐ are the diagonals.
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And when we find the area of a square, we usually say the area equals ๐ squared, one side length squared.
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Before we move on, I want us to think about the relationship between a square and a rhombus.
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But letโs start first with a parallelogram.
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One type of parallelogram is a rectangle.
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Another type of parallelogram is a rhombus.
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And a square is both a rectangle and a rhombus.
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And while all of this is really interesting information, for this problem, weโre only interested in this fact that a square is a rhombus.
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And that means that any rules that apply to a rhombus will also apply to a square, even the formula for finding the area.
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We know that a square has two diagonals.
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If we label one of them ๐ and one of them ๐, we can say that ๐ equals ๐.
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The diagonals are equal to one another.
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And then we can say that the area of a square can also be found by squaring the length of its diagonal and dividing by two.
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And this is true because a square is a rhombus.
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Now, we are ready to actually consider the problem in hand.
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We have a rhombus and a square.
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The rhombus has a diagonal of 3.8 centimeters and a diagonal of 5.3 centimeters.
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And so the area of this rhombus will be equal to 3.8 times 5.3 divided by two.
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Our square has a diagonal of 5.3 centimeters.
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And we know this means both diagonals will be equal to 5.3 centimeters.
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And so we can say that the area of the square will be equal to 5.3 squared divided by two.
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3.8 times 5.3 divided by two equals 10.07.
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Since this is area weโre dealing with centimeters squared, 5.3 squared divided by two equals 14.045 which again will be centimeters squared.
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Our question wants to know which shape is smaller in area.
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As 10.07 is smaller than 14.045, the rhombus is smaller in area than the square.