If 𝑊𝑋𝑌𝑍 is a kite, find 𝑍𝑌.
We’re told that the quadrilateral 𝑊𝑋𝑌𝑍 is a kite. What does this mean? Well, a kite is a quadrilateral with two pairs of consecutive congruent sides. In this question, it means that 𝑊𝑋 and 𝑊𝑍 are the same length and 𝑍𝑌 and 𝑋𝑌 are the same length. We’ve been given the length of two lines in the diagram: 𝑋𝑍 and 𝑊𝑌, which are the diagonals of the kite, as each connect a pair of opposite vertices. The length we’ve been asked to find is 𝑍𝑌, one of the longest sides of the kite. Let’s think about how to do this.
One of the key properties of a kite is that its diagonals are perpendicular. This means that the lines 𝑊𝑌 and 𝑋𝑍 are perpendicular. And hence, all four of the angles where they intersect are right angles. If we focus on the lower part of the diagram, we can now see that the line 𝑍𝑌 is part of a right-angled triangle — triangle 𝑃𝑌𝑍.
In this triangle, we know the length of two of the sides: they are seven and 17. And we’d like to calculate the length of the third side 𝑍𝑌. As the triangle is right angled, we can apply the Pythagorean theorem. 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. In this triangle, this means that 𝑍𝑌 squared is equal to seven squared plus 17 squared.
Now, we have an equation that we can solve in order to find the length of 𝑍𝑌. Evaluating seven squared and 17 squared gives 𝑍𝑌 squared is equal to 49 plus 289. Summing these two values tells us that 𝑍𝑌 squared is equal to 338. To find the value of 𝑍𝑌, we next need to square root. So we have that 𝑍𝑌 is equal to the square root of 338.
Now, this surd can be simplified. If we recognize that 338 has a square factor, it is equal to 169 multiplied by two. 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.
So we have that 𝑍𝑌 is equal to the square root of 169 multiplied by the square root of two. Remember 169 is a square number. So its square root is an integer. It’s 13. Therefore, we have that the length of 𝑍𝑌 as a simplified surd is 13 root two. Remember the key fact we used in this question was that if the quadrilateral is a kite, then its diagonals are perpendicular.