Midpoints and Distance in the Complex Plane (Grades 11-12)

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Midpoints and Distance in the Complex Plane

1. 
Let the graph below represent the complex plane, where x is the real axis and y is the imaginary axis. What is the midpoint of the line segment created by joining the two complex numbers represented by J and C?
Coordinate Plane - 5x5 - With Dots
  1. [math]1/2 + 5/2 i[/math]
  2. [math]3/2 + 3/2 i[/math]
  3. [math]-3/2 -3/2 i[/math]
  4. [math]1 + 5i[/math]
2. 
If [math]z= -11 + 2i[/math] and [math]w= -1 + 6i[/math] are plotted on the complex plane, and a line segment created by joining the two points, what would be the midpoint of this line segment?
  1. [math]-5 - 2i[/math]
  2. [math]-5/2 + 5/2i[/math]
  3. [math]-6 + 4i[/math]
  4. [math]-6 + 3i[/math]
3. 
If [math]-3 - 9i[/math] and [math]3 + i[/math] are plotted on the complex plane, and a line segment is created by joining these two points, what is the midpoint of this line segment?
  1. [math]-2 - 5i[/math]
  2. [math]-4i[/math]
  3. [math]2 + 5i[/math]
  4. [math]3/2 - 92 i[/math]
4. 
Which of the listed complex numbers, when represented by a point in the complex plane, is both equidistant and collinear with the complex numbers represented by points A and K?
Coordinate Plane - 10x10 - With Dots
  1. [math]3i[/math]
  2. [math]1 + 3/2 i[/math]
  3. [math]1/2 + 3i[/math]
  4. [math]3 + 1/2 i[/math]
5. 
Let [math]z=-8+5i[/math]. If the midpoint of the line segment created by complex numbers [math]z[/math] and [math]w[/math] is [math]-3+3/2 i[/math], what is the value of [math]w ?[/math]
  1. [math]w = -11/2 + 13/4 i[/math]
  2. [math]w = 2-2i[/math]
  3. [math]w = 14 + i[/math]
  4. [math]w = 4 + 7/2 i[/math]
6. 
Find the distance between [math]z = 6 + 7i[/math] and [math]w = -2 - 3i[/math] in the complex plane.
  1. [math]4sqrt(2)[/math]
  2. [math]18[/math]
  3. [math]2sqrt(2)[/math]
  4. [math]2sqrt(41)[/math]
7. 
Which of the following expressions represent(s) the distance between [math]z_1[/math] and [math]z_2[/math] in the complex plane? Choose all correct answers.
  1. [math]|z_1 - z_2|[/math]
  2. [math]|z_2 - z_1|[/math]
  3. [math]|z_2| - |z_1|[/math]
  4. [math]|z_2 + z_1|[/math]
8. 
What is the distance between the complex numbers [math]2-4i[/math] and [math]-3/2 - 1/2 i[/math] in the complex plane?
  1. [math]7/2 sqrt(2)[/math]
  2. [math]1/2sqrt(130)[/math]
  3. [math]5/2 sqrt(2)[/math]
  4. [math]1/8 sqrt(82)[/math]
9. 
If [math]z = 5 + 4i[/math], and is plotted in the complex plane, which of the following complex numbers would be 3 units away from [math]z[/math] in the complex plane? Choose all correct answers.
  1. [math]2 + 4i[/math]
  2. [math]6 + 6i[/math]
  3. [math]3 + (4+sqrt(5)) \ i[/math]
  4. [math]8 + i[/math]
10. 
Kayslee is given the complex numbers C and J in polar form, [math]2sqrt(5) \ (cos(63.4° + i sin63.4°)[/math] and [math] sqrt(2) \ (cos(135° + isin135°)[/math] respectively. She notices that she can find the distance between these numbers, [math]d[/math], in the complex plane simply by taking the difference of their arguments, [math]71.6°[/math], and then applying the law of cosines formula, [math] d = sqrt( (2sqrt(5))^2 + (sqrt(2))^2 - 2 * 2sqrt(5) * sqrt(2) cos71.6° )[/math]. Is this value correct? Check using the method for complex numbers in rectangular form. Is Kayslee's method applicable to any two complex numbers in the complex plane?
Coordinate Plane - 5x5 - With Dots
  1. Yes, the value is correct, and Kayslee's method will work for all complex numbers (being careful with how the difference of arguments is calculated).
  2. Yes, the value is correct, but this method only works in some instances.
  3. Yes, the value is correct, but this is merely coincidence (there is no reason for it).
  4. No, this value is not correct, and her method is also not correct (it does not correctly calculate the distance between complex numbers in the complex plane).

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