# 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?

- [math]1/2 + 5/2 i[/math]
- [math]3/2 + 3/2 i[/math]
- [math]-3/2 -3/2 i[/math]
- [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?

- [math]-5 - 2i[/math]
- [math]-5/2 + 5/2i[/math]
- [math]-6 + 4i[/math]
- [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?

- [math]-2 - 5i[/math]
- [math]-4i[/math]
- [math]2 + 5i[/math]
- [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?

- [math]3i[/math]
- [math]1 + 3/2 i[/math]
- [math]1/2 + 3i[/math]
- [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]

- [math]w = -11/2 + 13/4 i[/math]
- [math]w = 2-2i[/math]
- [math]w = 14 + i[/math]
- [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.

- [math]4sqrt(2)[/math]
- [math]18[/math]
- [math]2sqrt(2)[/math]
- [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.

- [math]|z_1 - z_2|[/math]
- [math]|z_2 - z_1|[/math]
- [math]|z_2| - |z_1|[/math]
- [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?

- [math]7/2 sqrt(2)[/math]
- [math]1/2sqrt(130)[/math]
- [math]5/2 sqrt(2)[/math]
- [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.

- [math]2 + 4i[/math]
- [math]6 + 6i[/math]
- [math]3 + (4+sqrt(5)) \ i[/math]
- [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?

- 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).
- Yes, the value is correct, but this method only works in some instances.
- Yes, the value is correct, but this is merely coincidence (there is no reason for it).
- 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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