Question

\(\mathrm{A}\left(\mathrm{z}_{1}\right), \mathrm{B}\left(\mathrm{z}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{z}_{3}\right)\) are vertices of \(\triangle \mathrm{ABC}\) where \(\mathrm{m}\) \(\angle \mathrm{C}=(\pi / 2)\) and \(\mathrm{AC}=\mathrm{BC}, \mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}\) are complex number if \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}=\mathrm{k}\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) then \(\mathrm{k}=\) (a) 1 (b) 2 (c) 4 (d) non of these

Short answer

The value of \(k\) is 2. Hence, the correct option is (b) 2.

Step-by-step solution

Understanding the Problem

We are given a triangle ABC where AC = BC and angle C is 90 degrees. This means ABC is an isosceles right triangle. We also know the complex values of points A, B, and C. The equation \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}=\mathrm{k}\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) is given and we are expected to value of k.

Using Complex Analysis Facts

We know from complex analysis that if AB = AC in triangle ABC, then \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2} = 2\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\), where z1, z2, and z3 are the complex representations of points A, B, and C respectively.

Substituting the Values

Substituting \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}\) for \(2\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) in the equation, we get the equality \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2} = k\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\).

Solving for k

Therefore, by comparing we get the value of k = 2. Hence the correct option is (b) 2. By following the proof, we've learned that the value of \( k \) is 2. If we compare this to the options, we can observe that option B is the correct choice.

Key concepts

Isosceles Right Triangle

An isosceles right triangle is a special type of triangle that features two sides of equal length, with the angle between them measuring 90 degrees. This gives the triangle one right angle and two equal angles of 45 degrees each. Imagine a triangle where the two legs are identical in length, making the hypotenuse the longest side.

Such a triangle arises from halving a square along its diagonal. Since both legs are equal, the diagonal splits both 90-degree angles of the square into equal 45-degree angles, forming the signature 45°, 45°, 90° angles.

• In our exercise, triangle ABC is an isosceles right triangle with AC = BC, and angle C = 90 degrees.
• This yields the identity that makes calculations simpler, as two sides are equal.

In a complex plane, the isosceles right triangle still adheres to these principles, with its vertices expressed as complex numbers. Understanding these properties helps us leverage geometric relations in calculations.

Complex Analysis

Complex analysis is a branch of mathematics exploring functions that use complex numbers, which include a real part and an imaginary part. Complex numbers are often written in the form of \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit.

In geometric problems involving triangles, complex numbers are a powerful tool to represent points in a plane. Each vertex of a triangle can be denoted as a complex number, simplifying many aspects of geometric transformation and distance computation.

• In our task, the vertices \( A(z_1), B(z_2), \) and \( C(z_3) \) of triangle ABC are given as complex numbers.
• When using complex analysis, we manipulate these numbers according to algebraic rules while interpreting geometrically in the 2D plane.

Applying complex analysis provides a unique perspective and toolkit for solving geometric problems, like determining the value of \( k \) in a given relation.

Triangle Geometry

Triangle geometry is a study focusing on the properties and characteristics of triangles, which are simple yet profound geometric shapes. In this realm, relationships between sides, angles, and various mathematical constructs are examined. A triangle has three sides and three angles that add up to 180 degrees.

For isosceles and isosceles right triangles, properties like equal angles and sides play a crucial role in geometric calculations.

• In our problem, the isosceles right triangle ensures that the sides AC and BC are identical.
• The equality of sides has specific implications for the calculation of the complex number expressions we used.

Using triangle geometry, we can derive important theorems and properties that assist significantly in problem-solving. In both plane and complex analysis geometry, understanding these properties is an essential skill, which allows students to navigate problems involving lengths, angles, and relations proficiently.