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If ππππ is a kite, find ππ.
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Weβre told that the quadrilateral ππππ is a kite.
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What does this mean?
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Well, a kite is a quadrilateral with two pairs of consecutive congruent sides.
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In this question, it means that ππ and ππ are the same length and ππ and ππ are the same length.
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Weβve been given the length of two lines in the diagram: ππ and ππ, which are the diagonals of the kite, as each connect a pair of opposite vertices.
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The length weβve been asked to find is ππ, one of the longest sides of the kite.
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Letβs think about how to do this.
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One of the key properties of a kite is that its diagonals are perpendicular.
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This means that the lines ππ and ππ are perpendicular.
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And hence, all four of the angles where they intersect are right angles.
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If we focus on the lower part of the diagram, we can now see that the line ππ is part of a right-angled triangle β triangle πππ.
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In this triangle, we know the length of two of the sides: they are seven and 17.
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And weβd like to calculate the length of the third side ππ.
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As the triangle is right angled, we can apply the Pythagorean theorem.
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Remember the Pythagorean theorem tells us that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
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In this triangle, this means that ππ squared is equal to seven squared plus 17 squared.
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Now, we have an equation that we can solve in order to find the length of ππ.
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Evaluating seven squared and 17 squared gives ππ squared is equal to 49 plus 289.
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Summing these two values tells us that ππ squared is equal to 338.
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To find the value of ππ, we next need to square root.
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So we have that ππ is equal to the square root of 338.
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Now, this surd can be simplified.
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If we recognize that 338 has a square factor, it is equal to 169 multiplied by two.
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The laws of surds tell us that we can separate out the square root of a product into the product of the individual square roots.
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So we have that ππ is equal to the square root of 169 multiplied by the square root of two.
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Remember 169 is a square number.
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So its square root is an integer.
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Itβs 13.
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Therefore, we have that the length of ππ as a simplified surd is 13 root two.
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Remember the key fact we used in this question was that if the quadrilateral is a kite, then its diagonals are perpendicular.