Complex Plane and Transformations Quiz

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Questions and Answers

What is the unit of measurement used for the graphic in the complex plane?

1 cm

What is the center and angle of rotation for r1?

Center: 0, Angle: π/3

What is the equation being solved in part A, question 1?

3y = 5(15-x)

What is the name of the circle being referenced in part A, question 2?

<p>Trigonometric circle ζ</p> Signup and view all the answers

What is the symbol used to represent the distance traveled by the point A on the circle ζ?

<p>d</p> Signup and view all the answers

What is the distance traveled by point A when it undergoes a rotation r1?

<p>π/3 cm</p> Signup and view all the answers

What is the goal of question 3 in Part A?

<p>To find all possible values of p and q where point A has travelled exactly two and a half times the circumference of circle ζ after p rotations of r1 and q rotations of r2.</p> Signup and view all the answers

What is the name of the homothety with center O and ratio 4?

<p>h1</p> Signup and view all the answers

What are the symbols used to represent the homothety of h1 and h2 on the circleζ?

<p>s1 and s2</p> Signup and view all the answers

What is the composition of s1, m being a natural number?

<p>S_m</p> Signup and view all the answers

What is the formula for calculating the composition of s1, m times, followed by s2, n times?

<p>f=S'_n o S_m</p> Signup and view all the answers

What is the center of the direct similitude 'f'?

<p>O</p> Signup and view all the answers

Flashcards

Complex Plane

A coordinate system where each point is represented by a complex number of the form a + bi, where 'a' is the real part and 'b' is the imaginary part.

Rotation r₁

A rotation of an angle π/3 radians around the origin O in the complex plane.

Rotation r₂

A rotation of an angle π/5 radians around the origin O in the complex plane.

Solution Set of Equation (E)

The set of all ordered pairs (x, y) that satisfy the equation 3y = 5(15 - x).

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Point I with affix 1

The point on the unit circle (circle with radius 1) in the complex plane.

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Distance Traveled (d)

The distance traveled by point A along the unit circle after undergoing p rotations r₁ and q rotations r₂.

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Homothety h₁

A dilation or resizing of the complex plane with a center O and a scale factor of 4.

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Homothety h₂

A dilation or resizing of the complex plane with a center O and a scale factor of -6.

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Similarity s₁

A combination of a rotation r₁ and a homothety h₁, represented as r₁oh₁.

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Similarity s₂

A combination of a rotation r₂ and a homothety h₂, represented as r₂oh₂.

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Composition sm

A repeated application of the similarity s₁ for 'm' times, expressed as s₁∘s₁∘...∘s₁.

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Composition sn

A repeated application of the similarity s₂ for 'n' times, expressed as s₂∘s₂∘...∘s₂.

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Transformation f

A combination of the composition sm and sn, expressed as sm∘sn.

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Direct Similarity

A transformation that preserves shape and orientation while changing size.

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Center of Similarity

The center point of the similarity transformation, where all points are mapped proportionally.

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Ratio of Similarity

The scaling factor of a similarity transformation.

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Angle of Similarity

The angle of rotation in a similarity transformation.

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Properties of Transformation f

The transformation 'f' is a direct similarity with a center O, a ratio of 22m x 3-3n, and an angle of mπ/3 + nπ/5.

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f is a Direct Similarity

The transformation 'f' is a direct similarity because it preserves shape and orientation while changing size.

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Study Notes

Complex Plane

  • Complex plane uses an orthonormal direct coordinate system (0,i,j) with a unit of 1 cm.
  • Rotation r₁: center O, angle π/3
  • Rotation r₂: center O, angle π/5

Part A

  • Equation: 3y = 5(15 - x) where x and y are integers.
  • Point I has an affix of 1.
  • Point A moves on the unit circle (ζ) centered at O, starting at I.
  • Distance A travels after p rotations of r₁ and q rotations of r₂ is calculated in cm.
  • Rotation r₁ covers a distance of π/3 cm.
  • Rotation r₂ covers a distance of π/5 cm.
  • Find integer values of p and q such that A travels exactly 2.5 times around the unit circle.

Part B

  • Homothétie h₁: center O, ratio 4
  • Homothétie h₂: center O, ratio -6
  • Define s₁ = r₁ ◦ h₁ and s₂ = r₂ ◦ h₂.
  • Find the nature and characteristics of s₁ and s₂.
  • Define composite s₁ ◦ s₁... ◦s₁ (m times) = Sm
  • Define composite s₂ ◦ s₂... ◦s₂ (n times) = Sn
  • Define f = Sm ◦ Sn
  • Prove f is a direct similarity with center O, ratio 2^2m × 3^n, and angle (mπ/3 + nπ/5)

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