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Chapter 3: Problem 159

If \(z_{1}^{2}+2 z_{2}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\), where \(z_{1}, z_{2}, z_{3}\) are the vertices of a triangle, then the triangle is (A) isosceles (B) right angled (C) equilateral (D) obtuse angled

Short Answer

Expert verified
The triangle is isosceles (Option A).

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation: \(z_{1}^{2} + 2z_{2}^{2} + z_{3}^{2} = 2z_{2}(z_{1} + z_{3})\). This can be rearranged as \(z_{1}^{2} + 2z_{2}^{2} + z_{3}^{2} - 2z_{2}z_{1} - 2z_{2}z_{3} = 0\).
02

Convert to Complex Number Form

Since \(z_{1}, z_{2}, z_{3}\) are vertices of a triangle in the complex plane, recognize that \(z_{1}, z_{2}, z_{3}\) correspond to complex numbers. The equation implies a relationship between these complex numbers.
03

Analyze Geometric Implications

Simplify the expression to resemble something geometric. Rewriting gives \((z_{1} - z_{2})^2 + (z_{3} - z_{2})^2 = 0\). This simplies to \((z_{1} - z_{2})^2 = -(z_{3} - z_{2})^2\).
04

Determine Triangle Type

The equation \((z_{1} - z_{2})^2 = -(z_{3} - z_{2})^2\) implies that the segments \(z_{1}z_{2}\) and \(z_{3}z_{2}\) are equal in length, because the squares are negatives of each other, which implies that the sides are conjugates and equal in triangle geometry context. Thus, the triangle is isosceles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Properties
In geometry, understanding triangle properties allows you to determine the kind of triangle you're dealing with based on its sides and angles. Three vertices define a triangle, and in the complex plane, these vertices can be represented as complex numbers. The properties of triangles can reveal much about their geometry:
  • Side Lengths: The length of sides in a triangle is paramount. For complex number vertices, the distance between two vertices is simply the modulus of their difference.

  • Angle Relationships: The sum of interior angles in any triangle is always 180 degrees.

  • Vertex Configuration: The geometric configuration of the vertices can involve symmetry.
Remember, using complex numbers to understand these properties provides a powerful tool in visualizing and solving geometric problems.
Complex Number Equations
Complex numbers are numbers that have both real and imaginary parts, and they can be quite useful in representing geometric figures like triangles. A complex number is typically noted as \(z = a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
  • Distance Calculation: In geometry, we often use the modulus \(|z|\) or absolute value to represent the distance between points, computed as \(\sqrt{a^2 + b^2}\).

  • Equation Formation: Relationships between vertices often translate into equations. For example, if given \(z_1^2 + 2z_2^2 + z_3^2 = 2z_2(z_1 + z_3)\), it signifies a particular symmetry or equality.

  • Transforms and Conjugates: Transformation through conjugates and other simple manipulations often simplifies complex geometrical problems.
Utilizing these aspects, one can deduce many properties of triangles by looking at complex number equations.
Isosceles Triangle
An isosceles triangle is defined by having at least two sides of equal length. This equal length results in equal angles opposite those sides, making it symmetric.
How do you detect an isosceles triangle when working with complex numbers?
  • Equation Symmetry: If an equation derived from triangle vertices expresses that two sides are equally long, the triangle is isosceles. In the context of the exercise, it was shown by rewriting and comparing terms, resulting in an equality indicating equal side lengths.

  • Real-World Applications: Identifying isosceles triangles using complex numbers can simplify solving related geometric problems or optimize designs in fields like engineering and architecture.
In geometry, understanding properties of special triangles like isosceles ones can greatly improve comprehension and provide insights into broader geometric applications.

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Most popular questions from this chapter

\(e^{2 m i \cot ^{-1} p} \cdot\left(\frac{p i+1}{p i-1}\right)^{m}=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(z_{1}+z_{2}+z_{3}=A, z_{1}+z_{2} \omega+z_{3} \omega^{2}=B\) and \(z_{1}+z_{2}\) \(\omega^{2}+z_{3} \omega=C\), where \(1, \omega, \omega^{2}\) are the three cube roots of unity, then \(|A|^{2}+|B|^{2}+|C|^{2}=\) (A) \(3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (B) \(2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (C) \(\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (D) None of these

If the complex numbers \(z_{1}, z_{2}, z_{3}\) are the vertices \(A\), \(B, C\) respectively of an isosceles right angled triangle with right angle at \(C\), then \(\left(z_{1}-z_{2}\right)^{2}=k\left(z_{1}-z_{3}\right)\left(z_{3}-z_{2}\right)\), where \(k=\) (A) 1 (B) 2 (C) 4 (D) None of these

The roots of the equation \(z^{4}-z^{3}+z^{2}-z+1=0\) are \(\cos \left(\frac{p \pi}{5}\right)+i \sin \left(\frac{p \pi}{5}\right)\) where \(p=\) (A) \(1,3,5,7,9\) (B) \(1,3,7,9\) (C) \(3,5,7,9\) (D) None of these

If \(|z-25 i| \leq 15\), then the least positive value of arg \(z\) is \(\begin{array}{ll}\text { (A) } \pi-\tan ^{-1} \frac{4}{3} & \text { (B) } \tan ^{-1} \frac{4}{3}\end{array}\) (C) \(-\pi+\tan ^{-1} \frac{4}{3}\) (D) None of these
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