Distance Midpoints in the Complex Plane - Algebra 2

1. On the complex plane, the point $z=a+bi$ corresponds to the coordinate $(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$ (the same as the coordinate geometry formulas). For $z_1=1-3i$ and $z_2=5+i$, what are the distance between them and the midpoint of the segment joining them?
   Answer: Distance $=\sqrt{32}$, midpoint $=3-i$ 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!
2. On the complex plane, the distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (same as the 2D distance formula), and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=2+5i$ and $z_2=6-1i$, what are the distance and the midpoint?
   Answer: Distance $=\sqrt{52}$, midpoint $=4+2i$ 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 = 2 + 5i and z₂ = 6 - i: (1) Calculate difference: (2 + 5i) - (6 - i) = -4 + 6i. (2) Find modulus: |−4 + 6i| = √((-4)² + 6²) = √(16 + 36) = √52. (3) For midpoint: (2 + 5i) + (6 - i) = 8 + 4i, then divide by 2 to get 4 + 2i. Choice A correctly applies both formulas to get distance = √52 and midpoint = 4 + 2i. Choice B incorrectly calculates the distance as √40, likely making an arithmetic error when computing 16 + 36 = 52, not 40! Choice D shows the difference 4 - 6i (with wrong sign) instead of the distance—remember that distance is always a non-negative real number, found by taking the modulus of the difference! These formulas connect complex arithmetic to geometry: the complex plane IS a coordinate system!
3. On the complex plane, let $z_1=1+7i$ and $z_2=5-1i$. Using $|z_1-z_2|$ for distance and $\dfrac{z_1+z_2}{2}$ for midpoint (the same as the coordinate formulas for $(a,b)$ and $(c,d)$), which option correctly gives both the distance and the midpoint?
   Answer: Distance $=\sqrt{80}$, midpoint $=3+3i$ 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—these formulas connect complex number arithmetic with geometry beautifully. To find the distance from 1 + 7i to 5 - i: (1) Calculate difference: (1 + 7i) - (5 - i) = -4 + 8i; (2) Find modulus: |-4 + 8i| = √(16 + 64) = √80; (3) Distance is √80 units—for the midpoint: (1) Add: (1 + 7i) + (5 - i) = 6 + 6i; (2) Divide by 2: (6 + 6i)/2 = 3 + 3i, 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 distance formula as the modulus of the difference and the midpoint formula as the average of the two complex numbers. Choice C calculates distance by just subtracting complex numbers without taking the modulus: the difference -4 + 8i (or 4 - 8i) 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. 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!
4. On the complex plane, $z=a+bi$ corresponds to $(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=0+0i$ and $z_2=-6+8i$, what are the distance and the midpoint?
   Answer: Distance $=10$, midpoint $=-3+4i$ 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 = 0 + 0i = 0 (the origin) and z₂ = -6 + 8i: (1) Calculate difference: (0 + 0i) - (-6 + 8i) = 6 - 8i. (2) Find modulus: |6 - 8i| = √(6² + (-8)²) = √(36 + 64) = √100 = 10. (3) For midpoint: (0 + 0i) + (-6 + 8i) = -6 + 8i, then divide by 2 to get -3 + 4i. Choice A correctly applies both formulas to get distance = 10 and midpoint = -3 + 4i. Choice B incorrectly calculates the distance as √28, making an error in the arithmetic. When one point is the origin, the distance to any point z = a + bi is simply |z| = √(a² + b²), which here gives √(36 + 64) = √100 = 10. The midpoint between origin and any point is exactly half that point!
5. Points $z_1=-1+4i$ and $z_2=5-4i$ lie on the complex plane. Using the geometric interpretation that $|z_1-z_2|$ is the length of the vector from $z_2$ to $z_1$ and equals $\sqrt{(a-c)^2+(b-d)^2}$, what is the distance between them?
   Answer: $10$ 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 is the modulus of their difference: distance = |z₁ - z₂| = √((a - c)² + (b - d)²), which represents the length of the vector from one point to the other. To find the distance from z₁ = -1 + 4i to z₂ = 5 - 4i: (1) Calculate the difference: (-1 + 4i) - (5 - 4i) = (-1 - 5) + (4 - (-4))i = -6 + 8i. (2) Find the modulus: |-6 + 8i| = √((-6)² + 8²) = √(36 + 64) = √100 = 10. The distance is 10 units. Choice A correctly shows 10 as the distance. Choice C shows √100, which equals 10 but isn't simplified - always simplify square roots when possible! The geometric interpretation is that we're finding the length of the straight line connecting these two points on the complex plane.
6. Points $z_1=4-i$ and $z_2=2+5i$ are plotted on the complex plane (so $a+bi$ corresponds to $(a,b)$). Using the midpoint formula $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$, what is the midpoint of the segment joining $z_1$ and $z_2$?
   Answer: $3+2i$ 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).
7. On the complex plane, $z=a+bi$ corresponds to $(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$. Let $z_1=3+2i$ and $z_2=-1+2i$. What are the distance and the midpoint?
   Answer: Distance $=4$, midpoint $=1+2i$ 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, averaging the complex numbers. For z₁ = 3 + 2i and z₂ = -1 + 2i: (1) Calculate difference: (3 + 2i) - (-1 + 2i) = 4 + 0i = 4. (2) Find modulus: |4 + 0i| = √(4² + 0²) = √16 = 4. (3) For midpoint: (3 + 2i) + (-1 + 2i) = 2 + 4i, then divide by 2 to get 1 + 2i. Choice A correctly applies both formulas to get distance = 4 and midpoint = 1 + 2i. Choice D gets the distance right but incorrectly writes the midpoint as just 1 (a real number) instead of 1 + 2i; when the imaginary parts of both complex numbers are equal (both are 2i here), their average is still 2i, not 0! The midpoint formula gives ((3 + (-1))/2) + ((2 + 2)/2)i = 1 + 2i. These formulas work because complex numbers form a 2D coordinate system where horizontal = real axis and vertical = imaginary axis!
8. Points $z_1=a+bi$ and $z_2=c+di$ on the complex plane are separated by distance $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, the same as the 2D distance formula. What is the distance between $z_1=5+2i$ and $z_2=1-3i$?
   Answer: $\sqrt{41}$ 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.
9. Complex numbers represent points: $z=a+bi\leftrightarrow(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=7-2i$ and $z_2=1+4i$, what are the distance and the midpoint?
   Answer: Distance $=\sqrt{72}$, midpoint $=4+i$ 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 = 7 - 2i and z₂ = 1 + 4i: (1) Calculate difference: (7 - 2i) - (1 + 4i) = 6 - 6i. (2) Find modulus: |6 - 6i| = √(6² + (-6)²) = √(36 + 36) = √72. (3) For midpoint: (7 - 2i) + (1 + 4i) = 8 + 2i, then divide by 2 to get 4 + i. Choice A correctly applies both formulas to get distance = √72 and midpoint = 4 + i. Choice B incorrectly calculates the distance as √36, perhaps computing only 6² = 36 and forgetting to add the (-6)² term, or making an arithmetic error. Remember: when finding |6 - 6i|, you need √(6² + 6²) = √(36 + 36) = √72, not √36! The complex plane formulas perfectly mirror coordinate geometry because complex numbers ARE coordinates!
10. Treat $z=a+bi$ as the point $(a,b)$ on the complex plane. The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$. For $z_1=-4+0i$ and $z_2=2+6i$, what are the distance and midpoint?
    Answer: Distance $=\sqrt{72}$, midpoint $=-1+3i$ 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 = -4 + 0i = -4 and z₂ = 2 + 6i: (1) Calculate difference: (-4 + 0i) - (2 + 6i) = -6 - 6i. (2) Find modulus: |−6 - 6i| = √((-6)² + (-6)²) = √(36 + 36) = √72. (3) For midpoint: (-4 + 0i) + (2 + 6i) = -2 + 6i, then divide by 2 to get -1 + 3i. Choice A correctly applies both formulas to get distance = √72 and midpoint = -1 + 3i. Choice B incorrectly calculates √36 instead of √72, perhaps forgetting to add both squared terms or making an arithmetic error: (-6)² + (-6)² = 36 + 36 = 72, not 36! Even when one complex number is purely real (like -4), the formulas work exactly the same way. The complex plane unifies real and imaginary numbers in one geometric framework!
11. A regular hexagon in the complex plane has its center at $2 + 3i$ and one vertex at $5 + 3i$. What is the distance between two adjacent vertices of this hexagon?
    Answer: $3$ The distance from the center $2 + 3i$ to the vertex $5 + 3i$ is the radius: $|(5 + 3i) - (2 + 3i)| = |3| = 3$. In a regular hexagon, the distance between adjacent vertices equals the radius. This is because the central angle between adjacent vertices is $60°$, and the triangle formed by the center and two adjacent vertices is equilateral. Therefore, the distance between adjacent vertices is 3. Choice B might result from incorrectly using $45°$ angles, choice C from using the side length formula for other regular polygons, and choice D from using the diameter instead of radius.
12. Points $z_1$ and $z_2$ on the complex plane form a line segment. The midpoint is $\dfrac{z_1+z_2}{2}$. What complex number is the midpoint of the segment joining $z_1=-6+4i$ and $z_2=2-2i$?
    Answer: $-2+i$ 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 + bi and z₂ = c + di is (z₁ + z₂)/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 -6 + 4i to 2 - 2i: (1) Add: (-6 + 4i) + (2 - 2i) = -4 + 2i. (2) Divide by 2: (-4 + 2i)/2 = -2 + i. The midpoint is at -2 + i, perfectly centered—nice job! Choice A correctly applies the midpoint formula as the average of the two complex numbers. Choice B adds without dividing properly or errs in signs: -4 + 2i is the sum, divide both parts by 2—don't forget the average! For a transferable strategy, add and divide by 2: for example, midpoint of -1 + 2i and 3 - 4i is ( -1+3 )/2 + (2-4)/2 i = 1 - i, just like averaging points. You're doing great— these concepts will click more with practice!
13. Let $z_1=-6+4i$ and $z_2=2-2i$ be points on the complex plane. Using the midpoint formula $\frac{z_1+z_2}{2}=\left(\frac{a+c}{2}\right)+\left(\frac{b+d}{2}\right)i,$ what is the midpoint of the segment joining $z_1$ and $z_2$?
    Answer: $-2+i$ 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.
14. Treat $z=a+bi$ as the point $(a,b)$ on the complex plane. Using the midpoint formula $\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i,$ what is the midpoint of the segment joining $z_1=-1+3i$ and $z_2=5+i$?
    Answer: $2+2i$ 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.
15. A complex number $z=a+bi$ represents the point $(a,b)$ on the complex plane. Using the coordinate-geometry distance formula, the distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=-2-i$ and $z_2=4+3i$, what are the distance and the midpoint?
    Answer: Distance $=\sqrt{52}$, midpoint $=1+i$ 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! The midpoint formula: midpoint = (z₁ + z₂)/2 = ((a + c)/2) + ((b + d)/2)i, averaging the real and imaginary parts separately. For z₁ = -2 - i and z₂ = 4 + 3i: (1) Calculate difference: (-2 - i) - (4 + 3i) = -6 - 4i. (2) Find modulus: |−6 - 4i| = √((-6)² + (-4)²) = √(36 + 16) = √52. (3) For midpoint: (-2 - i) + (4 + 3i) = 2 + 2i, then divide by 2 to get 1 + i. Choice A correctly applies both formulas to get distance = √52 and midpoint = 1 + i. Choice B incorrectly calculates the distance as √40 instead of √52, perhaps making an error when squaring -6 or -4, or when adding 36 + 16. Always double-check your arithmetic: (-6)² = 36 and (-4)² = 16, so 36 + 16 = 52, not 40! The beauty of these formulas is that they connect complex number arithmetic with geometry seamlessly—the complex plane truly is a coordinate system!
16. On the complex plane, the distance between $z_1=a+bi$ and $z_2=c+di$ equals the modulus of their difference: $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, and the midpoint is $\dfrac{z_1+z_2}{2}$. For $z_1=-1-5i$ and $z_2=5-1i$, what are the distance and the midpoint?
    Answer: Distance $=\sqrt{52}$, midpoint $=2-3i$ 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!
17. Let $z_1=5+2i$ and $z_2=1-3i$ be points on the complex plane. The distance between them is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$. What is the distance from $z_1$ to $z_2$?
    Answer: $\sqrt{41}$ 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!
18. Let $z_1=2+3i$ and $z_2=8+1i$ represent points on the complex plane. The midpoint is $\dfrac{z_1+z_2}{2}=\left(\dfrac{a+c}{2}\right)+\left(\dfrac{b+d}{2}\right)i$, matching the coordinate midpoint formula. What is the midpoint?
    Answer: $5+2i$ 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.
19. Let $z_1=-2-i$ and $z_2=3+2i$ represent points on the complex plane. Using $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (the same as the 2D distance formula), what is the distance from $z_1$ to $z_2$?
    Answer: $\sqrt{34}$ 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), exactly like the 2D distance formula! To find the distance from z₁ = -2 - i to z₂ = 3 + 2i: (1) Calculate the difference: (-2 - i) - (3 + 2i) = (-2 - 3) + (-1 - 2)i = -5 - 3i. (2) Find the modulus: |-5 - 3i| = √((-5)² + (-3)²) = √(25 + 9) = √34. The distance is √34 units. Choice A correctly shows √34 as the distance. Choice B shows √26, which might come from an arithmetic error like √(16 + 10) or √(25 + 1). When finding the modulus of -5 - 3i, square BOTH parts: (-5)² = 25 and (-3)² = 9, then add to get 34. Distance formula for complex numbers: subtract, find modulus of the result, and that's your distance - the formula is symmetric so order doesn't matter!
20. Complex numbers can be viewed as points $(a,b)$ on the complex plane. The midpoint of the segment from $z_1=a+bi$ to $z_2=c+di$ is $\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i$ which matches the coordinate midpoint formula. What is the midpoint of the segment from $z_1=4-i$ to $z_2=2+5i$?
    Answer: $3+2i$ 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 $z_1 = a + bi$ and $z_2 = c + di$ is $(z_1 + z_2)/2 = ((a + c)/2) + ((b + d)/2)i$, averaging the real and imaginary parts separately, just like in coordinate geometry! To find the midpoint from $4 - i$ to $2 + 5i$: (1) Add: $(4 - i) + (2 + 5i) = 6 + 4i$. (2) Divide by 2: $(6 + 4i)/2 = 3 + 2i$. The midpoint is at $3 + 2i$, 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: $6 + 4i$ 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 $1 + 3i$ and $5 - i$ is $(1 + 5)/2 + (3 - 1)/2 i = 3 + i$, averaging coordinates like in geometry. You're building strong skills here—keep going, and complex plane problems will feel intuitive!

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