Math Problem Statement

In the complex plane, solve geometric problems involving triangle properties, symmetry, and transformations as described in the uploaded image.

Solution

This image contains three problems in the complex plane related to geometry and transformations. I will break down and solve each part for clarity.


1. (Plan Complexe)

Given points B(1+i)B(1 + i) and C(2+2i)C(-2 + 2i):

a) Find the affix of AA such that ABC\triangle ABC is an isosceles right triangle with a direct orientation and AA as the right angle.

  1. The property of an isosceles right triangle is that the two equal sides are perpendicular.
  2. Let zA=x+yiz_A = x + yi. The condition is: zAzB=zAzC|z_A - z_B| = |z_A - z_C| and the angle between ABAB and ACAC must be 9090^\circ (direct orientation).

b) Construct ABC\triangle ABC.

Once zAz_A is determined in part (a), plot AA, BB, and CC in the complex plane.

c) Find the affix of EE, the symmetric of AA with respect to the midpoint of BCBC.

  1. Midpoint MM of BCBC: zM=zB+zC2z_M = \frac{z_B + z_C}{2}
  2. Symmetric point zEz_E: zE=2zMzAz_E = 2z_M - z_A

2. (Plan Complexe)

Given ABCABC is an isosceles right triangle in AA, and aa and bb are the affixes of AA and BB, respectively:

a) Find the affix of CC in terms of aa and bb.

Use the rotational property to find zCz_C such that: zC=zA+i(zBzA)z_C = z_A + i(z_B - z_A)

b) Find DD such that ABCDABCD is a parallelogram.

For DD: zD=zC+(zBzA)z_D = z_C + (z_B - z_A)


3. (Plan Complexe)

Given points A(zA=2)A(z_A = 2) and B(zB=1+3i)B(z_B = 1 + \sqrt{3}i), with: Z=zBzAZ = \frac{z_B}{z_A}

a) Express ZZ in exponential form.

Convert ZZ into polar coordinates: Z=zBzA=1+3i2Z = \frac{z_B}{z_A} = \frac{1 + \sqrt{3}i}{2}

b) Deduce the nature of triangle OABOAB.

Analyze the angles and magnitudes to determine if OAB\triangle OAB is isosceles, equilateral, or right-angled.


Would you like detailed solutions to any specific part of these problems? Let me know! Here are 5 questions to expand your understanding:

  1. How do we interpret geometric properties in the complex plane using affixes?
  2. What role does symmetry play in transformations of complex numbers?
  3. How do rotations in the complex plane relate to the exponential form of complex numbers?
  4. What is the geometric interpretation of the modulus and argument of a complex number?
  5. How can we verify triangle properties using distance and angles in the complex plane?

Tip: When working with isosceles right triangles in the complex plane, remember that rotations by 9090^\circ involve multiplying by ii.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Geometry in the Complex Plane
Symmetry
Rotations and Transformations

Formulas

Distance formula: |z1 - z2|
Midpoint formula: (z1 + z2) / 2
Rotation by 90°: Multiplication by i
Symmetry: z_symmetric = 2z_mid - z_original
Polar form of complex numbers: z = r(cosθ + i sinθ)

Theorems

Properties of Isosceles Right Triangles
Geometric Transformations Using Complex Numbers
Relationship Between Argument and Modulus

Suitable Grade Level

Grades 10-12