Question Video: Geometry in the Complex Plane Mathematics

Video Transcript

A complex number 𝑤 lies at a distance of five root two from 𝑧 one equals three plus five 𝑖 and at a distance of four root five from 𝑧 two equals negative six minus two 𝑖. Is the triangle formed by the points 𝑤, 𝑧 one, and 𝑧 two a right triangle?

So we have 𝑧 one equals three plus five 𝑖. And 𝑧 two equals negative six minus two 𝑖, which we can mark accurately on our Argand diagram or complex plane. However, it’s hard to guess where the complex number 𝑤 should go. All we know is that it lies at the distance of five root two from 𝑧 one and four root five from 𝑧 two. The question is whether the triangle with these vertices is a right triangle. And as we know, two of the lengths are suggested using the Pythagorean theorem.

If the square of the length of the longest side equals the sum of the squares of the other two sides, then this is a right triangle. But first, we need to find this longest side length, which we’ll call 𝑑. 𝑑 is the distance between the complex numbers 𝑧 one and 𝑧 two. And so it’s the modulus of 𝑧 one minus 𝑧 two. We substitute the known values of 𝑧 one and 𝑧 two, subtract the complex numbers to get nine plus seven 𝑖. Its modulus is the square root of nine squared plus seven squared, which is the square root of 130.

Now, we can apply the converse of the Pythagorean theorem. We need to identify the longest side then. Remember, our diagram might not be that accurate. We can unsimplify the other two side lengths to get root 50 and root 80, respectively. And so the length of the longest side really is 𝑑. We just need to check then whether 𝑑 squared equals the sum of squares of the other two side lengths. As 𝑑 is root 130, 𝑑 squared is 130. Five root two squared is five squared, which is 25 times two. And similarly, four root five squared is four squared, which is 16 times five. And it’s 130 equals 50 plus 80. Yes, it is. And so our triangle is a right triangle, with the right angle at 𝑤.