Algebra 2 Flashcards: Distance Midpoints In The Complex Plane

Study Distance Midpoints In The Complex Plane in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra 2

Distance Midpoints In The Complex Plane

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QUESTION

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

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ANSWER

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

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What this deck covers

This deck focuses on Distance Midpoints In The Complex Plane, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer:

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

Algebra 2 Flashcards: Distance Midpoints In The Complex Plane

Distance Midpoints In The Complex Plane

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Find the midpoint of z1=−3iz_1=-3iz1​=−3i and z2=9iz_2=9iz2​=9i.

Flashcard 2: Find the distance between z1=1+2iz_1=1+2iz1​=1+2i and z2=4+6iz_2=4+6iz2​=4+6i.

Flashcard 3: Find the midpoint of z1=−2−2iz_1=-2-2iz1​=−2−2i and z2=4+6iz_2=4+6iz2​=4+6i.

Flashcard 4: Find the midpoint of z1=3−5iz_1=3-5iz1​=3−5i and z2=3+7iz_2=3+7iz2​=3+7i.

Flashcard 5: Find the distance between z1=3+0iz_1=3+0iz1​=3+0i and z2=0+4iz_2=0+4iz2​=0+4i.

Flashcard 6: Find ∣z∣|z|∣z∣ for z=5−12iz=5-12iz=5−12i.

Flashcard 7: What is the distance between z1=a+biz_1=a+biz1​=a+bi and z2=c+diz_2=c+diz2​=c+di in coordinates?

Flashcard 8: Find the distance between z1=−6+8iz_1=-6+8iz1​=−6+8i and z2=0z_2=0z2​=0.

Flashcard 9: Find and correct the formula error: distance claimed as ∣z1+z2∣|z_1+z_2|∣z1​+z2​∣ for endpoints z1z_1z1​ and z2z_2z2​.

Flashcard 10: Find ∣z1−z2∣|z_1-z_2|∣z1​−z2​∣ for z1=7−3iz_1=7-3iz1​=7−3i and z2=1−3iz_2=1-3iz2​=1−3i.

Flashcard 11: Find the distance between z1=−3+0iz_1=-3+0iz1​=−3+0i and z2=5+0iz_2=5+0iz2​=5+0i.

Flashcard 12: Find the midpoint of z1=2−4iz_1=2-4iz1​=2−4i and z2=−6+0iz_2=-6+0iz2​=−6+0i.

Flashcard 13: What is the modulus of z=a+biz=a+biz=a+bi written in terms of aaa and bbb?

Flashcard 14: Identify the distance between z1z_1z1​ and z2z_2z2​ if z1−z2=−9+0iz_1-z_2=-9+0iz1​−z2​=−9+0i.

Flashcard 15: What is the midpoint mmm if z2=z1+wz_2=z_1+wz2​=z1​+w for a complex step www?

Flashcard 16: Find the midpoint of z1=−4+2iz_1=-4+2iz1​=−4+2i and z2=6+8iz_2=6+8iz2​=6+8i.

Flashcard 17: Find and correct the formula error: midpoint claimed as z1−z22\frac{z_1-z_2}{2}2z1​−z2​​ for endpoints z1z_1z1​ and z2z_2z2​.

Flashcard 18: Find ∣z1−z2∣|z_1-z_2|∣z1​−z2​∣ for z1=4+iz_1=4+iz1​=4+i and z2=1+5iz_2=1+5iz2​=1+5i.

Flashcard 19: Find the distance between z1=0+2iz_1=0+2iz1​=0+2i and z2=0−4iz_2=0-4iz2​=0−4i.

Flashcard 20: Find the midpoint of z1=−7+3iz_1=-7+3iz1​=−7+3i and z2=1−5iz_2=1-5iz2​=1−5i.

Flashcard 21: Find the midpoint of z1=−8z_1=-8z1​=−8 and z2=2z_2=2z2​=2.

Flashcard 22: Find the distance between z1=2−4iz_1=2-4iz1​=2−4i and z2=−6+0iz_2=-6+0iz2​=−6+0i.

Flashcard 23: What is ∣z1−z2∣|z_1-z_2|∣z1​−z2​∣ interpreted as geometrically in the complex plane?

Flashcard 24: State the formula for the distance between complex numbers z1z_1z1​ and z2z_2z2​ in the complex plane.

Flashcard 25: What is the midpoint of z1=az_1=az1​=a and z2=bz_2=bz2​=b when both endpoints are real numbers?

Flashcard 26: Find the midpoint of z1=2+6iz_1=2+6iz1​=2+6i and z2=8+0iz_2=8+0iz2​=8+0i.

Flashcard 27: Find the midpoint of z1=1+2iz_1=1+2iz1​=1+2i and z2=9+10iz_2=9+10iz2​=9+10i.

Flashcard 28: Find the distance between z1=5z_1=5z1​=5 and z2=−1+4iz_2=-1+4iz2​=−1+4i.

Flashcard 29: Identify the real part of the midpoint m=z1+z22m=\frac{z_1+z_2}{2}m=2z1​+z2​​ for z1=a+biz_1=a+biz1​=a+bi and z2=c+diz_2=c+diz2​=c+di.

Flashcard 30: Find ∣z∣|z|∣z∣ for z=9+12iz=9+12iz=9+12i.

Flashcard 31: Find the distance between z1=−1+2iz_1=-1+2iz1​=−1+2i and z2=2−2iz_2=2-2iz2​=2−2i.

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

Flashcard 33: Find the distance between z1=3+4iz_1=3+4iz1​=3+4i and z2=0z_2=0z2​=0.

Flashcard 34: Find the distance between z1=−8z_1=-8z1​=−8 and z2=2z_2=2z2​=2.

Flashcard 35: Find the distance between z1=1+iz_1=1+iz1​=1+i and z2=1−iz_2=1-iz2​=1−i.

Flashcard 36: What is the distance between z1=az_1=az1​=a and z2=bz_2=bz2​=b when both endpoints are real numbers?

Flashcard 37: Identify the imaginary part of the midpoint m=z1+z22m=\frac{z_1+z_2}{2}m=2z1​+z2​​ for z1=a+biz_1=a+biz1​=a+bi and z2=c+diz_2=c+diz2​=c+di.

Flashcard 38: Find the distance between z1=−3iz_1=-3iz1​=−3i and z2=9iz_2=9iz2​=9i.

Flashcard 39: What is the distance between z1z_1z1​ and z2z_2z2​ if z1−z2=3−4iz_1-z_2=3-4iz1​−z2​=3−4i?

Flashcard 40: Find the distance between z1=−2+1iz_1=-2+1iz1​=−2+1i and z2=4−7iz_2=4-7iz2​=4−7i.

Flashcard 41: Identify the distance between z1z_1z1​ and z2z_2z2​ if z1−z2=0+7iz_1-z_2=0+7iz1​−z2​=0+7i.

Flashcard 42: What is the distance from the origin to zzz in the complex plane, written using modulus?

Flashcard 43: Find the midpoint of z1=3+0iz_1=3+0iz1​=3+0i and z2=0+4iz_2=0+4iz2​=0+4i.

Flashcard 44: Find the midpoint of z1=0z_1=0z1​=0 and z2=6+8iz_2=6+8iz2​=6+8i.

Flashcard 45: Find the distance between z1=2+3iz_1=2+3iz1​=2+3i and z2=5+3iz_2=5+3iz2​=5+3i.

Flashcard 46: Find the midpoint of z1=5z_1=5z1​=5 and z2=−1+4iz_2=-1+4iz2​=−1+4i.

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

Flashcard 48: What is the distance between z1z_1z1​ and z2z_2z2​ if z2=z1+(6+8i)z_2=z_1+(6+8i)z2​=z1​+(6+8i)?

Flashcard 49: State the formula for the midpoint of endpoints z1z_1z1​ and z2z_2z2​ in the complex plane.

Flashcard 50: What is the distance between z1z_1z1​ and z2z_2z2​ if z1−z2=−8+6iz_1-z_2=-8+6iz1​−z2​=−8+6i?

Flashcard 51: What is the midpoint between z1z_1z1​ and z2z_2z2​ interpreted as geometrically in the complex plane?

Flashcard 52: Find the distance between z1=−2+5iz_1=-2+5iz1​=−2+5i and z2=−2−1iz_2=-2-1iz2​=−2−1i.