Understanding the Triangle Formed by Points Representing Complex Numbers

To determine the nature of the triangle formed by the points representing the given complex numbers, we first need to treat these complex numbers as points in the Argand plane. A complex number \(x + yi\) corresponds to the point \((x, y)\) in the Cartesian coordinate system.

The given complex numbers are:

• \(z_1 = 3 + 4i\), corresponding to point \(A(3, 4)\).
• \(z_2 = 8 - 6i\), corresponding to point \(B(8, -6)\).
• \(z_3 = 13 + 9i\), corresponding to point \(C(13, 9)\).

The nature of the triangle formed by these points depends on the lengths of its sides. The distance between two points representing complex numbers \(z_a\) and \(z_b\) is given by the magnitude of their difference, i.e., \(|z_a - z_b|\). This is equivalent to using the distance formula between the corresponding Cartesian coordinates.

Calculating Side Lengths of the Triangle

Let's calculate the lengths of the three sides of the triangle formed by these points using the distance formula or the magnitude of the difference between the complex numbers:

Side AB: Distance between \(z_1\) and \(z_2\).

\(AB = |z_2 - z_1| = |(8 - 6i) - (3 + 4i)| = |(8 - 3) + (-6 - 4)i| = |5 - 10i|\)

\(AB = \sqrt{5^2 + (-10)^2} = \sqrt{25 + 100} = \sqrt{125}\)

\(AB^2 = 125\)

Side BC: Distance between \(z_2\) and \(z_3\).

\(BC = |z_3 - z_2| = |(13 + 9i) - (8 - 6i)| = |(13 - 8) + (9 - (-6))i| = |5 + 15i|\)

\(BC = \sqrt{5^2 + 15^2} = \sqrt{25 + 225} = \sqrt{250}\)

\(BC^2 = 250\)

Side AC: Distance between \(z_1\) and \(z_3\).

\(AC = |z_3 - z_1| = |(13 + 9i) - (3 + 4i)| = |(13 - 3) + (9 - 4)i| = |10 + 5i|\)

\(AC = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125}\)

\(AC^2 = 125\)

The side lengths squared are \(AB^2 = 125\), \(BC^2 = 250\), and \(AC^2 = 125\).

Determining the Nature of the Triangle

Now we examine the relationship between the side lengths to determine the nature of the triangle formed by these complex numbers.

• Since \(AB^2 = 125\) and \(AC^2 = 125\), we have \(AB = AC\). This indicates that the triangle is at least an isosceles triangle.
• Let's check if the Pythagorean theorem holds for these side lengths: \(a^2 + b^2 = c^2\).
  • Is \(AB^2 + AC^2 = BC^2\)? \(125 + 125 = 250\). Yes, \(250 = 250\).

Since the sum of the squares of two sides (\(AB^2\) and \(AC^2\)) is equal to the square of the third side (\(BC^2\)), the triangle satisfies the Pythagorean theorem. This proves that the triangle formed by the points representing the complex numbers \(3+4i\), \(8-6i\), and \(13+9i\) is a right-angled triangle, with the right angle at the vertex corresponding to the complex number that is common to sides AB and AC, which is \(z_1\) (or point A).

The distance formula and the concept of complex numbers help us determine the type of triangle formed by points in the complex plane. The triangle formed by these points is a right-angled triangle.

Comparing the lengths squared, we see \(AB^2 + AC^2 = 125 + 125 = 250 = BC^2\). Thus, the triangle formed is a right angled triangle.