A kite has vertices at the points two, zero; three, two; four, zero; and three, negative three. Work out the perimeter of the kite. Give your solution to one decimal place. Work out the area of the kite.
As the coordinates given here are easy to plot, let’s start by drawing a diagram of this kite. Now, we’ve plotted our coordinates and joined them up. And we can see that it is indeed a kite since a kite has two pairs of consecutive sides of equal length. Here, our top two sides would be the same length and the bottom two sides would be the same length. In this question, the first part is asking us to find the perimeter of the kite. And we remember that the perimeter of a shape is the distance around the outside edge. So we need to know the lengths of all the outside edges. We don’t currently know the lengths of these sides. But let’s see if we can work them out using the diagonals of our kite.
If we use the fact that the diagonals of a kite are perpendicular, then this means that we have formed four triangles within our kite that have 90-degree angles. So this means if we call the length of one of our top sides 𝑥, we can use the Pythagorean theorem to help us find it. The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides of the triangle. So for our top triangle, we know that our hypotenuse is 𝑥. The vertical length must be two units and the horizontal length is one unit.
So using the Pythagorean theorem, we would have 𝑥 squared equals one squared plus two squared. And it doesn’t matter which way round we put our one and two. Simplifying this, we would have 𝑥 squared equals one plus four since two squared is the same as two times two. So 𝑥 squared is equal to five. And to find 𝑥 by itself, we would take the inverse operation, which is the square root of both sides. So 𝑥 is equal to square root of five. And we can leave our answer in this root form for the next part of our workings.
Returning to our kite diagram, let’s define the length of one of the lower sides as 𝑦. In this case, our hypotenuse is 𝑦. Our horizontal length will also be one. And our vertical length will be three units. Applying the Pythagorean theorem would give us 𝑦 squared equals three squared plus one squared. So 𝑦 squared equals nine plus one, which is 10. To find 𝑦 then, we take the square root of both sides, giving us that 𝑦 equals root 10. Returning then to the question of finding the perimeter of this kite, using the properties of the kite, we know that we have a length 𝑥 on top. So the other consecutive length must also be 𝑥. The same is true for the lower lengths. We know that one side is defined as 𝑦. So equally, the other length must also be 𝑦.
The perimeter then must be 𝑥 plus 𝑥 plus 𝑦 plus 𝑦 or simplified as two 𝑥 plus two 𝑦. As we worked out that 𝑥 is equal to root five and 𝑦 was equal to root 10, we can substitute these in to give us the perimeter equals two root five plus two root 10. Using our calculator, we can evaluate this as 10.79669128 and so on. But we’re asked to give a solution to one decimal place. Checking our second decimal place to see if it’s five or more, we see that this is a nine, which means that our answer will run up to 10.8.
So now, let’s move on to the second part of the question, finding the area of the kite. We could use a number of different methods to do this. We could use the fact that there are four triangles in our kite and work out the area of each of these four triangles and add them together. However, a slightly faster method is to use the fact that the area of a kite equals 𝑝 times 𝑞 over two, where 𝑝 and 𝑞 are the lengths of the diagonals. In our kite, the length of the horizontal diagonal would be two units and the length of the vertical diagonal would be five units.
So the area of our kite is equal to two times five — that’s our two diagonal lengths — divided by two. Simplifying this, we get 10 over two which is five. So our final answers are the perimeter is 10.8 and the area of the kite is five.