Video Transcript
If ππππ is a kite, find ππ.
Weβre told that the quadrilateral ππππ 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 ππ and ππ are the same length and ππ and ππ are the same length. 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. The length weβve been asked to find is ππ, one of the longest sides of the kite. Letβs 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 ππ and ππ 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 ππ is part of a right-angled triangle β triangle πππ.
In this triangle, we know the length of two of the sides: they are seven and 17. And weβd like to calculate the length of the third side ππ. 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 ππ 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 ππ. Evaluating seven squared and 17 squared gives ππ squared is equal to 49 plus 289. Summing these two values tells us that ππ squared is equal to 338. To find the value of ππ, we next need to square root. So we have that ππ 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 ππ 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. Itβs 13. Therefore, we have that the length of ππ 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.
