### Video Transcript

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

So we have π§ one equals three plus
five π. And π§ two equals negative six
minus two π, which we can mark accurately on our Argand diagram or complex
plane. However, itβs hard to guess where
the complex number π€ should go. All we know is that it lies at the
distance of five root two from π§ one and four root five from π§ 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 weβll call π. π is the distance between the
complex numbers π§ one and π§ two. And so itβs the modulus of π§ one
minus π§ two. We substitute the known values of
π§ one and π§ two, subtract the complex numbers to get nine plus seven π. 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 π. We just need to check then whether
π squared equals the sum of squares of the other two side lengths. As π is root 130, π 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 itβs 130 equals 50 plus 80. Yes, it is. And so our triangle is a right
triangle, with the right angle at π€.