Algebra 2 Quiz: Distance Midpoints In The Complex Plane
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Distance Midpoints In The Complex PlaneQuestion 1 of 20

Let z1=−6+4iz_1=-6+4i and z2=2−2iz_2=2-2i be points on the complex plane. Using the midpoint formula z1+z22=(a+c2)+(b+d2)i,\frac{z_1+z_2}{2}=\left(\frac{a+c}{2}\right)+\left(\frac{b+d}{2}\right)i, what is the midpoint of the segment joining z1z_1 and z2z_2?

−2+i-2+i
−4+2i-4+2i
−2+3i-2+3i
−8+2i-8+2i
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Algebra 2 Quiz

Algebra 2 Quiz: Distance Midpoints In The Complex Plane

Practice Distance Midpoints In The Complex Plane in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Distance Midpoints In The Complex Plane, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let z1=−6+4iz_1=-6+4i and z2=2−2iz_2=2-2i be points on the complex plane. Using the midpoint formula z1+z22=(a+c2)+(b+d2)i,\frac{z_1+z_2}{2}=\left(\frac{a+c}{2}\right)+\left(\frac{b+d}{2}\right)i, what is the midpoint of the segment joining z1z_1 and z2z_2?

  1. −2+i-2+i (correct answer)
  2. −4+2i-4+2i
  3. −2+3i-2+3i
  4. −8+2i-8+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula is: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i. To find the midpoint of z₁ = -6 + 4i and z₂ = 2 - 2i: (1) Add the complex numbers: (-6 + 4i) + (2 - 2i) = -4 + 2i. (2) Divide by 2: (-4 + 2i)/2 = -2 + i. The midpoint is at -2 + i, exactly halfway between the two points. Choice A correctly applies the midpoint formula. Choice B shows -4 + 2i, which is the sum without dividing by 2—remember the averaging step! Choice D shows -8 + 2i, which might come from multiplying instead of adding. Always add the complex numbers first, THEN divide both real and imaginary parts by 2. The midpoint formula works because you're essentially averaging the x-coordinates (real parts) and y-coordinates (imaginary parts) separately.

Question 2

Treat z=a+biz=a+bi as the point (a,b)(a,b) on the complex plane. Using the midpoint formula z1+z22=a+c2+b+d2i,\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i, what is the midpoint of the segment joining z1=−1+3iz_1=-1+3i and z2=5+iz_2=5+i?

  1. 2+2i2+2i (correct answer)
  2. 4+4i4+4i
  3. 2+i2+i
  4. −3+2i-3+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula for complex numbers is midpoint = (z₁ + z₂)/2 = [(a + c)/2] + [(b + d)/2]i, averaging the real and imaginary parts separately. To find the midpoint of -1 + 3i and 5 + i: (1) Add: (-1 + 3i) + (5 + i) = 4 + 4i. (2) Divide by 2: (4 + 4i)/2 = 4/2 + 4i/2 = 2 + 2i. The midpoint is at 2 + 2i, equidistant from both original points! Choice A correctly applies the midpoint formula. Choice B shows 4 + 4i, the sum without dividing—always remember to divide by 2 to get the midpoint, not just the sum! The midpoint formula essentially averages the x-coordinates (real parts) and y-coordinates (imaginary parts) separately, just like in coordinate geometry.

Question 3

Points z1=a+biz_1=a+bi and z2=c+diz_2=c+di on the complex plane are separated by distance ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}, the same as the 2D distance formula. What is the distance between z1=5+2iz_1=5+2i and z2=1−3iz_2=1-3i?

  1. 9\sqrt{9}
  2. 20\sqrt{20}
  3. 41\sqrt{41} (correct answer)
  4. 4−5i4-5i
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between z₁ = a + bi and z₂ = c + di is |z₁ - z₂| = √[(a - c)² + (b - d)²], exactly like the 2D distance formula. To find the distance from 5 + 2i to 1 - 3i: (1) Calculate difference: (5 + 2i) - (1 - 3i) = 5 + 2i - 1 + 3i = 4 + 5i. (2) Find modulus: |4 + 5i| = √(4² + 5²) = √(16 + 25) = √41. Choice A correctly applies the distance formula to get √41. Choice C shows 4 - 5i, which appears to be the difference with a sign error on the imaginary part—when subtracting 1 - 3i, remember that -(-3i) = +3i, so 2i - (-3i) = 2i + 3i = 5i! Always be careful with signs when subtracting complex numbers, especially when the second number has negative parts.

Question 4

Let z1=2+3iz_1=2+3i and z2=8+1iz_2=8+1i represent points on the complex plane. The midpoint is z1+z22=(a+c2)+(b+d2)i\dfrac{z_1+z_2}{2}=\left(\dfrac{a+c}{2}\right)+\left(\dfrac{b+d}{2}\right)i, matching the coordinate midpoint formula. What is the midpoint?

  1. 5+2i5+2i (correct answer)
  2. 10+4i10+4i
  3. 3+1i3+1i
  4. 5+1i5+1i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula averages the coordinates: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i, matching the coordinate midpoint formula exactly! To find the midpoint of z₁ = 2 + 3i and z₂ = 8 + 1i: (1) Add the complex numbers: (2 + 3i) + (8 + 1i) = (2 + 8) + (3 + 1)i = 10 + 4i. (2) Divide by 2: (10 + 4i)/2 = 10/2 + 4i/2 = 5 + 2i. The midpoint is at 5 + 2i, perfectly centered between the two points! Choice A correctly shows 5 + 2i as the midpoint. Choice B shows 10 + 4i, which is the sum before dividing - that's a common error where students forget the averaging step! The complex plane midpoint formula gives us the exact center of the line segment connecting two complex numbers.

Question 5

On the complex plane, the point z=a+biz=a+bi corresponds to (a,b)(a,b). The distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di is |z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2, the same as the coordinate distance formula. What is the distance between z1=3+2iz_1=3+2i and z2=1−iz_2=1-i?

  1. 5\sqrt{5}
  2. 2+3i2+3i
  3. 13\sqrt{13} (correct answer)
  4. 25\sqrt{25}
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z₁ = a + bi and z₂ = c + di is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²), which is exactly the 2D distance formula because the complex plane is a coordinate system! To find the distance from 3 + 2i to 1 - i: (1) Calculate difference: (3 + 2i) - (1 - i) = 2 + 3i. (2) Find modulus: |2 + 3i| = √(2² + 3²) = √(4 + 9) = √13. (3) The distance is √13 units—great job applying the formula! Choice C correctly applies the distance formula as the modulus of the difference. Choice B calculates the difference without taking the modulus: 2 + 3i is a vector, not the distance—you must find its length to get the scalar distance! For transferable strategy, always subtract the complex numbers, then take the modulus of the result: for example, distance from 4 + i to 2 - 3i is |(4 + i) - (2 - 3i)| = |2 + 4i| = √(4 + 16) = √20 = 2√5, symmetric regardless of subtraction order. Keep practicing these, and you'll master visualizing complex numbers geometrically—you've got this!

Question 6

On the complex plane, the point z=a+biz=a+bi corresponds to (a,b)(a,b). The distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di is ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2} (the same as the coordinate geometry distance formula). What is the distance between z1=3+2iz_1=3+2i and z2=1−iz_2=1-i?

  1. 2+3i2+3i
  2. 13\sqrt{13} (correct answer)
  3. 55
  4. 5\sqrt{5}
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di is the modulus of their difference: distance = ∣z1−z2∣=(a−c)2+(b−d)2|z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2}, which is exactly the 2D distance formula because the complex plane is a coordinate system! The midpoint formula also works the same: midpoint = (z1+z2)/2=((a+c)/2)+((b+d)/2)i(z_1 + z_2)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real and imaginary parts separately—these formulas connect complex number arithmetic with geometry beautifully. To find the distance from 3+2i3 + 2i to 1−i1 - i: (1) Calculate difference: (3+2i)−(1−i)=2+3i(3 + 2i) - (1 - i) = 2 + 3i; (2) Find modulus: ∣2+3i∣=22+32=4+9=13|2 + 3i| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}; (3) So the distance is 13\sqrt{13} units—geometrically, plot both points and the distance is the straight-line length between them! Choice B correctly applies the distance formula as the modulus of the difference. Choice D calculates the difference without taking the modulus: 2+3i2 + 3i is a vector, not the distance—you must find its length (modulus) to get the actual distance, so always take the square root after squaring and adding! Remember the distance formula for complex numbers: (1) Subtract the complex numbers (order doesn't matter for distance); (2) Find the modulus of the result: if the difference is p+qip + qi, modulus = p2+q2\sqrt{p^2 + q^2}; (3) That's the distance—keep practicing, you've got this! For example, distance from 4+i4 + i to 2−i2 - i: difference = 2+2i2 + 2i, modulus = 4+4=8=22\sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}, and it's symmetric if you subtract the other way.

Question 7

On the complex plane, z=a+biz=a+bi corresponds to the point (a,b)(a,b). For z1=−1+4iz_1=-1+4i and z2=3z_2=3, which complex number is the midpoint of the segment connecting them, using z1+z22\dfrac{z_1+z_2}{2}?

  1. 1+2i1+2i (correct answer)
  2. 2+4i2+4i
  3. 1+4i1+4i
  4. −2+2i-2+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula works the same: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real parts and imaginary parts separately—these formulas connect complex number arithmetic with geometry beautifully. For the midpoint of -1 + 4i and 3 + 0i: (1) Add: (-1 + 4i) + (3 + 0i) = 2 + 4i; (2) Divide by 2: (2 + 4i)/2 = 1 + 2i, exactly halfway between the two points on the complex plane—geometrically, plot both points and the midpoint lies on the line segment connecting them, equidistant from each! Choice A correctly applies the midpoint formula as the average of the two complex numbers. Choice B makes an arithmetic error by adding instead of averaging: (-1 + 4i) + (3 + 0i) = 2 + 4i, but you must divide the entire result by 2—midpoint formula: add both complex numbers first, then divide by 2, averaging both real and imaginary parts. Midpoint formula: (1) Add the two complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i; (2) Divide result by 2: real part (a + c)/2, imaginary part (b + d)/2; (3) Write as ((a + c)/2) + ((b + d)/2)i—essentially, average x-coordinates for real part, average y-coordinates for imaginary part, just like coordinate midpoint! For 4 + 2i and 2 - 6i: add to get 6 - 4i, divide by 2 to get 3 - 2i as midpoint—the complex plane formulas are identical to coordinate geometry because the complex plane is a coordinate system, and great job tackling these!

Question 8

On the complex plane, the difference z1−z2z_1-z_2 represents the vector from z2z_2 to z1z_1, and its modulus gives the distance: ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}. What is the distance between z1=2+3iz_1=2+3i and z2=−1+7iz_2=-1+7i?

  1. 7\sqrt{7}
  2. 25\sqrt{25} (correct answer)
  3. 3+4i3+4i
  4. 49\sqrt{49}
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance is |z₁ - z₂| = √[(a - c)² + (b - d)²]. To find the distance from 2 + 3i to -1 + 7i: (1) Calculate difference: (2 + 3i) - (-1 + 7i) = 2 + 3i + 1 - 7i = 3 - 4i. (2) Find modulus: |3 - 4i| = √(3² + (-4)²) = √(9 + 16) = √25 = 5. Choice B correctly gives √25. Choice C shows 3 + 4i, which has the wrong sign on the imaginary part—when subtracting -1 + 7i, we get 3i - 7i = -4i, not +4i! The distance formula gives the same result regardless of which point you subtract from which, as long as you correctly find the modulus of the difference.

Question 9

Complex numbers can be treated like points: z=a+bi↔(a,b)z=a+bi\leftrightarrow(a,b). The distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di is the modulus ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}, and the midpoint is z1+z22\dfrac{z_1+z_2}{2}. For z1=−3+1iz_1=-3+1i and z2=1−3iz_2=1-3i, what are the distance and the midpoint?

  1. Distance =32=\sqrt{32}, midpoint =−1−i=-1-i (correct answer)
  2. Distance =16=\sqrt{16}, midpoint =−1−i=-1-i
  3. Distance =32=\sqrt{32}, midpoint =−2+2i=-2+2i
  4. Distance =4−4i=4-4i, midpoint =−1−i=-1-i
Explanation: This question tests your understanding of finding distances and midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z₁ = a + bi and z₂ = c + di is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²). The midpoint formula: midpoint = (z₁ + z₂)/2. For z₁ = -3 + i and z₂ = 1 - 3i: (1) Calculate difference: (-3 + i) - (1 - 3i) = -4 + 4i. (2) Find modulus: |−4 + 4i| = √((-4)² + 4²) = √(16 + 16) = √32. (3) For midpoint: (-3 + i) + (1 - 3i) = -2 - 2i, then divide by 2 to get -1 - i. Choice A correctly applies both formulas to get distance = √32 and midpoint = -1 - i. Choice D shows a common misconception by writing the distance as 4 - 4i, which is just the difference of the complex numbers, not the distance! The distance must be a non-negative real number (the modulus), not a complex number. Always remember: subtract first to get the difference, then take the modulus to get the actual distance! The complex plane beautifully unifies algebra and geometry through these familiar formulas!

Question 10

Points z1=4−iz_1=4-i and z2=2+5iz_2=2+5i are plotted on the complex plane (so a+bia+bi corresponds to (a,b)(a,b)). Using the midpoint formula z1+z22=a+c2+b+d2i\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i, what is the midpoint of the segment joining z1z_1 and z2z_2?

  1. 3+2i3+2i (correct answer)
  2. 6+4i6+4i
  3. 1+2i1+2i
  4. 3−2i3-2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula for complex numbers works the same as in coordinate geometry: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real parts and imaginary parts separately. To find the midpoint of z₁ = 4 - i and z₂ = 2 + 5i: (1) Add the complex numbers: (4 - i) + (2 + 5i) = (4 + 2) + (-1 + 5)i = 6 + 4i. (2) Divide by 2: (6 + 4i)/2 = 6/2 + 4i/2 = 3 + 2i. The midpoint is at 3 + 2i, exactly halfway between the two points on the complex plane! Choice A correctly shows 3 + 2i as the midpoint. Choice B shows 6 + 4i, which is the sum of the two complex numbers before dividing by 2 - remember to complete the averaging step! The complex plane formulas are identical to coordinate geometry because the complex plane IS a coordinate system where a + bi corresponds to the point (a, b).

Question 11

On the complex plane, the point z=a+biz=a+bi corresponds to the coordinate (a,b)(a,b). The distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di is ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}, and the midpoint is z1+z22=a+c2+b+d2i\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i (the same as the coordinate geometry formulas). For z1=1−3iz_1=1-3i and z2=5+iz_2=5+i, what are the distance between them and the midpoint of the segment joining them?

  1. Distance =52=\sqrt{52}, midpoint =3−i=3-i
  2. Distance =32=\sqrt{32}, midpoint =3−i=3-i (correct answer)
  3. Distance =52=\sqrt{52}, midpoint =3−2i=3-2i
  4. Distance =4+4i=4+4i, midpoint =3−i=3-i
Explanation: This question tests your understanding of finding distances and midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z₁ = a + bi and z₂ = c + di is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²), which is exactly the 2D distance formula because the complex plane IS a coordinate system! The midpoint formula also works the same: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real parts and imaginary parts separately. For z₁ = 1 - 3i and z₂ = 5 + i: (1) Calculate difference: (1 - 3i) - (5 + i) = -4 - 4i. (2) Find modulus: |−4 - 4i| = √((-4)² + (-4)²) = √(16 + 16) = √32. (3) For midpoint: (1 - 3i) + (5 + i) = 6 - 2i, then divide by 2 to get 3 - i. Choice B correctly applies both formulas to get distance = √32 and midpoint = 3 - i. Choice A incorrectly calculates the distance as √52 instead of √32, likely making an arithmetic error when squaring -4 or adding the squares. Remember to carefully square each component: (-4)² = 16, not some other value! The complex plane formulas are identical to coordinate geometry because the complex plane IS a coordinate system where real part = x-coordinate and imaginary part = y-coordinate!

Question 12

Complex numbers correspond to points (a,b)(a,b) in the plane. Using the midpoint formula z1+z22=a+c2+b+d2i\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i, find the midpoint of the segment from z1=7−3iz_1=7-3i to z2=−1+5iz_2=-1+5i.

  1. 3+i3+i (correct answer)
  2. 6+2i6+2i
  3. 3−4i3-4i
  4. 8+2i8+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint of two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di is z1+z22=a+c2+b+d2i\frac{z_1 + z_2}{2} = \frac{a + c}{2} + \frac{b + d}{2}i, averaging the real and imaginary parts separately, just like in coordinate geometry! To find the midpoint from 7−3i7 - 3i to −1+5i-1 + 5i: (1) Add: (7−3i)+(−1+5i)=6+2i(7 - 3i) + (-1 + 5i) = 6 + 2i. (2) Divide by 2: 6+2i2=3+i\frac{6 + 2i}{2} = 3 + i. The midpoint is at 3+i3 + i—fantastic! Choice A correctly applies the midpoint formula as the average. Choice B adds without dividing: 6+2i6 + 2i is the sum, not the midpoint—always divide by 2! For a transferable strategy, average reals and imaginaries: midpoint of 2+4i2 + 4i and 6−2i6 - 2i is 2+62+4−22i=4+i\frac{2+6}{2} + \frac{4-2}{2} i = 4 + i, simple as that. You're progressing wonderfully—keep practicing!

Question 13

Let z1=5+2iz_1=5+2i and z2=1−3iz_2=1-3i be points on the complex plane. The distance between them is ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}. What is the distance from z1z_1 to z2z_2?

  1. 25\sqrt{25}
  2. 41\sqrt{41} (correct answer)
  3. 32\sqrt{32}
  4. 4−5i4-5i
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z₁ = a + bi and z₂ = c + di is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²), which is exactly the 2D distance formula because the complex plane is a coordinate system—these formulas connect complex number arithmetic with geometry beautifully. To find the distance from 5 + 2i to 1 - 3i: (1) Calculate difference: (5 + 2i) - (1 - 3i) = 4 + 5i; (2) Find modulus: |4 + 5i| = √(16 + 25) = √41; (3) Distance is √41 units—geometrically, plot both points and the distance is the straight-line length connecting them! Choice B correctly applies the distance formula as the modulus of the difference. Choice D calculates by just subtracting complex numbers without taking the modulus: the difference 4 + 5i is a vector, not a distance—you must find its length (modulus) to get the distance, so always take the modulus of the difference, don't stop at subtraction! Distance formula for complex numbers: (1) Subtract the complex numbers (order doesn't matter for distance): z₁ - z₂; (2) Find modulus of result: if difference is p + qi, modulus = √(p² + q²); (3) That's the distance—example: distance from 2 - 3i to 5 + i: difference = (2 - 3i) - (5 + i) = -3 - 4i, modulus = √(9 + 16) = 5, and the formula is symmetric—same answer if you subtract in the opposite order. You're making excellent progress—keep visualizing the points to reinforce the concepts!

Question 14

On the complex plane, the midpoint of the segment from z1=1−7iz_1=1-7i to z2=5+3iz_2=5+3i is found using z1+z22=(a+c2)+(b+d2)i.\frac{z_1 + z_2}{2} = \left(\frac{a + c}{2}\right) + \left(\frac{b + d}{2}\right)i. What is the midpoint?

  1. 2−5i2-5i
  2. 6−4i6-4i
  3. 3−2i3-2i (correct answer)
  4. 3+2i3+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula is: midpoint=z1+z22=(a+c2)+(b+d2)i\text{midpoint} = \frac{z_1 + z_2}{2} = \left( \frac{a + c}{2} \right) + \left( \frac{b + d}{2} \right)i. To find the midpoint of z1=1−7iz_1 = 1 - 7i and z2=5+3iz_2 = 5 + 3i: (1) Add the complex numbers: (1−7i)+(5+3i)=6−4i(1 - 7i) + (5 + 3i) = 6 - 4i. (2) Divide by 2: (6−4i)/2=3−2i(6 - 4i)/2 = 3 - 2i. The midpoint is at 3−2i3 - 2i, exactly halfway between the two points. Choice A correctly applies the midpoint formula. Choice B shows 6−4i6 - 4i, which is the sum without dividing by 2—don't forget the averaging step! Choice C shows 2−5i2 - 5i, which doesn't match our calculation. The midpoint formula essentially averages the x-coordinates (real parts) and y-coordinates (imaginary parts) separately, just like in regular coordinate geometry.

Question 15

On the complex plane, the distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di equals the modulus of their difference: ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}, and the midpoint is z1+z22\dfrac{z_1+z_2}{2}. For z1=−1−5iz_1=-1-5i and z2=5−1iz_2=5-1i, what are the distance and the midpoint?

  1. Distance =52=\sqrt{52}, midpoint =2−3i=2-3i (correct answer)
  2. Distance =40=\sqrt{40}, midpoint =2−3i=2-3i
  3. Distance =52=\sqrt{52}, midpoint =4−6i=4-6i
  4. Distance =6+4i=6+4i, midpoint =2−3i=2-3i
Explanation: This question tests your understanding of finding distances and midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between two complex numbers z₁ = a + bi and z₂ = c + di is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²). The midpoint formula: midpoint = (z₁ + z₂)/2. For z₁ = -1 - 5i and z₂ = 5 - i: (1) Calculate difference: (-1 - 5i) - (5 - i) = -6 - 4i. (2) Find modulus: |−6 - 4i| = √((-6)² + (-4)²) = √(36 + 16) = √52. (3) For midpoint: (-1 - 5i) + (5 - i) = 4 - 6i, then divide by 2 to get 2 - 3i. Choice A correctly applies both formulas to get distance = √52 and midpoint = 2 - 3i. Choice D shows 6 + 4i as the distance, which is just a complex number (and has wrong signs), not the modulus! Distance must be a non-negative real number. Always remember the process: subtract to get difference, then find modulus for distance. These formulas beautifully connect complex arithmetic with geometry!

Question 16

Complex numbers can be viewed as points (a,b)(a,b) on the complex plane. The midpoint of the segment from z1=a+biz_1=a+bi to z2=c+diz_2=c+di is z1+z22=a+c2+b+d2i,\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i, which matches the coordinate midpoint formula. What is the midpoint of the segment joining z1=4−iz_1=4-i and z2=2+5iz_2=2+5i?

  1. 3+2i3+2i (correct answer)
  2. 6+4i6+4i
  3. 1+2i1+2i
  4. 3−3i3-3i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint of two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di is (z1+z2)/2=((a+c)/2)+((b+d)/2)i(z_1 + z_2)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real and imaginary parts separately—this matches the coordinate midpoint formula perfectly because the complex plane is a coordinate system! The distance formula is similar: ∣z1−z2∣=(a−c)2+(b−d)2|z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2}, connecting arithmetic to geometry in such a cool way. To find the midpoint of 4−i4 - i and 2+5i2 + 5i: (1) Add: (4−i)+(2+5i)=6+4i(4 - i) + (2 + 5i) = 6 + 4i; (2) Divide by 2: 6/2+4i/2=3+2i6/2 + 4i/2 = 3 + 2i; the midpoint is at 3+2i3 + 2i, exactly halfway between the points on the plane—geometrically, it lies on the line segment connecting them, equidistant from each! Choice A correctly applies the midpoint formula as the average of the two complex numbers. Choice B adds without dividing: 6+4i6 + 4i is the sum, not the average—you must divide by 2 to find the midpoint, so don't forget that step! Master the midpoint formula: (1) Add the two complex numbers: (a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i; (2) Divide by 2: ((a+c)/2)+((b+d)/2)i((a + c)/2) + ((b + d)/2)i; (3) That's it—essentially averaging x-coordinates for real parts and y-coordinates for imaginary parts, just like in coordinate geometry! For example, midpoint of 1+3i1 + 3i and 5−i5 - i: sum = 6+2i6 + 2i, divide by 2 = 3+i3 + i—great job, you're building strong skills!

Question 17

Complex numbers can be viewed as points (a,b)(a,b) on the complex plane. The midpoint of the segment from z1=a+biz_1=a+bi to z2=c+diz_2=c+di is z1+z22=a+c2+b+d2i\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i which matches the coordinate midpoint formula. What is the midpoint of the segment from z1=4−iz_1=4-i to z2=2+5iz_2=2+5i?

  1. 3+2i3+2i (correct answer)
  2. 6+4i6+4i
  3. 3−3i3-3i
  4. 1+2i1+2i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint of two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di is (z1+z2)/2=((a+c)/2)+((b+d)/2)i(z_1 + z_2)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real and imaginary parts separately, just like in coordinate geometry! To find the midpoint from 4−i4 - i to 2+5i2 + 5i: (1) Add: (4−i)+(2+5i)=6+4i(4 - i) + (2 + 5i) = 6 + 4i. (2) Divide by 2: (6+4i)/2=3+2i(6 + 4i)/2 = 3 + 2i. The midpoint is at 3+2i3 + 2i, right in the center of the line segment—excellent work! Choice A correctly applies the midpoint formula as the average of the two complex numbers. Choice B adds without dividing: 6+4i6 + 4i is the sum, not the average—remember to divide by 2 to find the midpoint! For a transferable strategy, add the two complex numbers and divide by 2: for example, midpoint of 1+3i1 + 3i and 5−i5 - i is (1+5)/2+(3−1)/2i=3+i(1 + 5)/2 + (3 - 1)/2 i = 3 + i, averaging coordinates like in geometry. You're building strong skills here—keep going, and complex plane problems will feel intuitive!

Question 18

Points z1=4−iz_1=4-i and z2=2+5iz_2=2+5i are plotted on the complex plane (so a+bia+bi corresponds to (a,b)(a,b)). Using the midpoint formula z1+z22=a+c2+b+d2i\frac{z_1 + z_2}{2} = \frac{a + c}{2} + \frac{b + d}{2} i, what complex number is the midpoint of the segment from z1z_1 to z2z_2?

  1. 2+3i2+3i
  2. 3+2i3+2i (correct answer)
  3. 6+4i6+4i
  4. 3+4i3+4i
Explanation: This question tests your understanding of finding midpoints between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The midpoint formula for complex numbers works the same as in coordinate geometry: midpoint = (z1+z2)/2=((a+c)/2)+((b+d)/2)i(z_1 + z_2)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real parts and imaginary parts separately. To find the midpoint of z1=4−iz_1 = 4 - i and z2=2+5iz_2 = 2 + 5i: (1) Add the complex numbers: (4−i)+(2+5i)=6+4i(4 - i) + (2 + 5i) = 6 + 4i. (2) Divide by 2: (6+4i)/2=3+2i(6 + 4i)/2 = 3 + 2i. The midpoint is at 3+2i3 + 2i, exactly halfway between the two points on the complex plane. Choice A correctly applies the midpoint formula as the average of the two complex numbers. Choice B shows 6+4i6 + 4i, which is the sum without dividing by 2—remember to complete the averaging step! Always add first, THEN divide both real and imaginary parts by 2. Geometrically, if you plot both points, the midpoint lies on the line segment connecting them, equidistant from each!

Question 19

The modulus ∣z1−z2∣|z_1-z_2| gives the distance between complex numbers z1=a+biz_1=a+bi and z2=c+diz_2=c+di, just like the distance between (a,b)(a,b) and (c,d)(c,d) in coordinate geometry. What is the distance between z1=−3+4iz_1=-3+4i and z2=1+1iz_2=1+1i?

  1. 7\sqrt{7}
  2. 25\sqrt{25} (correct answer)
  3. 16\sqrt{16}
  4. −4+3i-4+3i
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance is |z₁ - z₂| = √[(a - c)² + (b - d)²]. To find the distance from -3 + 4i to 1 + 1i: (1) Calculate difference: (-3 + 4i) - (1 + i) = -3 + 4i - 1 - i = -4 + 3i. (2) Find modulus: |-4 + 3i| = √[(-4)² + 3²] = √(16 + 9) = √25 = 5. Choice B correctly gives √25 (which equals 5). Choice D shows -4 + 3i, which is the difference vector itself—remember that distance is the modulus (length) of this vector, not the vector itself! The modulus calculation squares both components, so negative signs don't affect the final positive distance value.

Question 20

Because complex numbers correspond to points (a,b)(a,b), the distance between z1=a+biz_1=a+bi and z2=c+diz_2=c+di is ∣z1−z2∣=(a−c)2+(b−d)2|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}. What is the distance between z1=−1+2iz_1=-1+2i and z2=2−2iz_2=2-2i?

  1. 7\sqrt{7}
  2. 25\sqrt{25}
  3. 3−4i3-4i
  4. 55 (correct answer)
Explanation: This question tests your understanding of finding distances between complex numbers on the complex plane using formulas that mirror 2D coordinate geometry. The distance between z₁ = a + bi and z₂ = c + di is |z₁ - z₂| = √((a - c)² + (b - d)²), which corresponds to the 2D distance formula! To find the distance from z₁ = -1 + 2i to z₂ = 2 - 2i: (1) Calculate the difference: (-1 + 2i) - (2 - 2i) = -1 + 2i - 2 + 2i = -3 + 4i. (2) Find the modulus: |-3 + 4i| = √((-3)² + 4²) = √(9 + 16) = √25 = 5. The distance is 5 units. Choice D correctly shows 5 as the distance (note that √25 = 5). Choice B (√25) is technically correct but not simplified—since √25 = 5, the simplified answer is 5. Choice C (3 - 4i) appears to show a difference without taking the modulus, though with incorrect signs. The distance formula gives us a non-negative real number, never a complex number! When you get √25, always simplify to 5 unless specifically asked to leave it as a square root. The distance between any two complex numbers is always a non-negative real number representing the length of the line segment connecting them on the complex plane.