Distance Midpoints in the Complex Plane - Algebra 2

1. Find the midpoint of $z_1=-3i$ and $z_2=9i$.

   $3i$. $\frac{-3i+9i}{2} = \frac{6i}{2} = 3i$.

2. Find the distance between $z_1=1+2i$ and $z_2=4+6i$.

   $5$. $|(1+2i)-(4+6i)| = |-3-4i| = \sqrt{9+16} = 5$.

3. Find the midpoint of $z_1=-2-2i$ and $z_2=4+6i$.

   $1+2i$. $\frac{(-2-2i)+(4+6i)}{2} = \frac{2+4i}{2} = 1+2i$.

4. Find the midpoint of $z_1=3-5i$ and $z_2=3+7i$.

   $3+i$. $\frac{(3-5i)+(3+7i)}{2} = \frac{6+2i}{2} = 3+i$.

5. Find the distance between $z_1=3+0i$ and $z_2=0+4i$.

   $5$. $|(3+0i)-(0+4i)| = |3-4i| = \sqrt{9+16} = 5$.

6. Find $|z|$ for $z=5-12i$.

   $13$. $|5-12i| = \sqrt{25+144} = \sqrt{169} = 13$.

7. What is the distance between $z_1=a+bi$ and $z_2=c+di$ in coordinates?

   $\sqrt{(a-c)^2+(b-d)^2}$. Apply the standard distance formula using coordinates $ (a,b) $ and $ (c,d) $.

8. Find the distance between $z_1=-6+8i$ and $z_2=0$.

   $10$. Distance from origin: $|-6+8i| = \sqrt{36+64} = 10$.

9. Find and correct the formula error: distance claimed as $|z_1+z_2|$ for endpoints $z_1$ and $z_2$.

   Correct distance: $|z_1-z_2|$. Distance requires subtraction, not addition of the complex numbers.

10. Find $|z_1-z_2|$ for $z_1=7-3i$ and $z_2=1-3i$.

    $6$. $|(7-3i)-(1-3i)| = |6| = 6$.

11. Find the distance between $z_1=-3+0i$ and $z_2=5+0i$.

    $8$. $|(-3+0i)-(5+0i)| = |-8| = 8$.

12. Find the midpoint of $z_1=2-4i$ and $z_2=-6+0i$.

    $-2-2i$. $\frac{(2-4i)+(-6+0i)}{2} = \frac{-4-4i}{2} = -2-2i$.

13. What is the modulus of $z=a+bi$ written in terms of $a$ and $b$?

    $|z|=\sqrt{a^2+b^2}$. Modulus uses the Pythagorean theorem with real and imaginary parts.

14. Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=-9+0i$.

    $9$. $|-9+0i| = |-9| = 9$.

15. What is the midpoint $m$ if $z_2=z_1+w$ for a complex step $w$?

    $m=z_1+\frac{w}{2}$. The midpoint is the starting point plus half the step vector.

16. Find the midpoint of $z_1=-4+2i$ and $z_2=6+8i$.

    $1+5i$. $\frac{(-4+2i)+(6+8i)}{2} = \frac{2+10i}{2} = 1+5i$.

17. Find and correct the formula error: midpoint claimed as $\frac{z_1-z_2}{2}$ for endpoints $z_1$ and $z_2$.

    Correct midpoint: $\frac{z_1+z_2}{2}$. Midpoint requires addition, not subtraction of the complex numbers.

18. Find $|z_1-z_2|$ for $z_1=4+i$ and $z_2=1+5i$.

    $5$. $|(4+i)-(1+5i)| = |3-4i| = \sqrt{9+16} = 5$.

19. Find the distance between $z_1=0+2i$ and $z_2=0-4i$.

    $6$. $|(0+2i)-(0-4i)| = |6i| = 6$.

20. Find the midpoint of $z_1=-7+3i$ and $z_2=1-5i$.

    $-3-i$. $\frac{(-7+3i)+(1-5i)}{2} = \frac{-6-2i}{2} = -3-i$.

21. Find the midpoint of $z_1=-8$ and $z_2=2$.

    $-3$. $\frac{-8+2}{2} = \frac{-6}{2} = -3$.

22. Find the distance between $z_1=2-4i$ and $z_2=-6+0i$.

    $\sqrt{80}$. $|(2-4i)-(-6+0i)| = |8-4i| = \sqrt{64+16} = \sqrt{80}$.

23. What is $|z_1-z_2|$ interpreted as geometrically in the complex plane?

    The distance between $z_1$ and $z_2$. The modulus of a difference represents geometric distance.

24. State the formula for the distance between complex numbers $z_1$ and $z_2$ in the complex plane.

    $d=|z_1-z_2|$. Distance is the modulus of the difference between two complex numbers.

25. What is the midpoint of $z_1=a$ and $z_2=b$ when both endpoints are real numbers?

    $\frac{a+b}{2}$. For real numbers, the midpoint formula reduces to the average.

26. Find the midpoint of $z_1=2+6i$ and $z_2=8+0i$.

    $5+3i$. $\frac{(2+6i)+(8+0i)}{2} = \frac{10+6i}{2} = 5+3i$.

27. Find the midpoint of $z_1=1+2i$ and $z_2=9+10i$.

    $5+6i$. $\frac{(1+2i)+(9+10i)}{2} = \frac{10+12i}{2} = 5+6i$.

28. Find the distance between $z_1=5$ and $z_2=-1+4i$.

    $\sqrt{52}$. $|5-(-1+4i)| = |6-4i| = \sqrt{36+16} = \sqrt{52}$.

29. Identify the real part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.

    $\frac{a+c}{2}$. Average the real parts: $\frac{a+c}{2}$.

30. Find $|z|$ for $z=9+12i$.

    $15$. $|9+12i| = \sqrt{81+144} = \sqrt{225} = 15$.

31. Find the distance between $z_1=-1+2i$ and $z_2=2-2i$.

    $5$. $|(-1+2i)-(2-2i)| = |-3+4i| = \sqrt{9+16} = 5$.

32. Identify the complex number that represents the midpoint of points $(a,b)$ and $(c,d)$.

    $\frac{a+c}{2}+\frac{b+d}{2}i$. Average the real parts and imaginary parts separately.

33. Find the distance between $z_1=3+4i$ and $z_2=0$.

    $5$. Distance from origin: $|3+4i| = \sqrt{3^2+4^2} = 5$.

34. Find the distance between $z_1=-8$ and $z_2=2$.

    $10$. $|(-8)-(2)| = |-10| = 10$

35. Find the distance between $z_1=1+i$ and $z_2=1-i$.

    $2$. $|(1+i)-(1-i)| = |2i| = 2$.

36. What is the distance between $z_1=a$ and $z_2=b$ when both endpoints are real numbers?

    $|a-b|$. For real numbers, distance is the absolute value of the difference.

37. Identify the imaginary part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.

    $\frac{b+d}{2}$. Average the imaginary parts: $\frac{b+d}{2}$.

38. Find the distance between $z_1=-3i$ and $z_2=9i$.

    $12$. $|(-3i)-(9i)| = |-12i| = 12$.

39. What is the distance between $z_1$ and $z_2$ if $z_1-z_2=3-4i$?

    $5$. $|3-4i| = \sqrt{9+16} = 5$.

40. Find the distance between $z_1=-2+1i$ and $z_2=4-7i$.

    $10$. $|(-2+1i)-(4-7i)| = |-6+8i| = \sqrt{36+64} = 10$.

41. Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=0+7i$.

    $7$. $|0+7i| = |7i| = 7$.

42. What is the distance from the origin to $z$ in the complex plane, written using modulus?

    $|z|$. The modulus gives the distance from any point to the origin.

43. Find the midpoint of $z_1=3+0i$ and $z_2=0+4i$.

    $\frac{3}{2}+2i$. $\frac{(3+0i)+(0+4i)}{2} = \frac{3+4i}{2} = \frac{3}{2}+2i$.

44. Find the midpoint of $z_1=0$ and $z_2=6+8i$.

    $3+4i$. $\frac{0+(6+8i)}{2} = \frac{6+8i}{2} = 3+4i$.

45. Find the distance between $z_1=2+3i$ and $z_2=5+3i$.

    $3$. $|(2+3i)-(5+3i)| = |-3| = 3$.

46. Find the midpoint of $z_1=5$ and $z_2=-1+4i$.

    $2+2i$. $\frac{5+(-1+4i)}{2} = \frac{4+4i}{2} = 2+2i$.

47. Identify the complex number that represents the midpoint of points $ (a,b) $ and $ (c,d) $.

    $\frac{a+c}{2}+\frac{b+d}{2}i$. Average the real parts and imaginary parts separately.

48. What is the distance between $z_1$ and $z_2$ if $z_2=z_1+(6+8i)$?

    $10$. $|6+8i| = \sqrt{36+64} = 10$.

49. State the formula for the midpoint of endpoints $z_1$ and $z_2$ in the complex plane.

    $m=\frac{z_1+z_2}{2}$. Average the two endpoints by adding and dividing by 2.

50. What is the distance between $z_1$ and $z_2$ if $z_1-z_2=-8+6i$?

    $10$. $|-8+6i| = \sqrt{64+36} = 10$.

51. What is the midpoint between $z_1$ and $z_2$ interpreted as geometrically in the complex plane?

    The point halfway along the segment from $z_1$ to $z_2$. The midpoint is equidistant from both endpoints on the line segment.

52. Find the distance between $z_1=-2+5i$ and $z_2=-2-1i$.

    6. $|(-2+5i)-(-2-1i)| = |6i| = 6$

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