Distance Midpoints in the Complex Plane - Algebra 2
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Find the distance between $z_1=-2+5i$ and $z_2=-2-1i$.
Find the distance between $z_1=-2+5i$ and $z_2=-2-1i$.
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- $|(-2+5i)-(-2-1i)| = |6i| = 6$
- $|(-2+5i)-(-2-1i)| = |6i| = 6$
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Find the midpoint of $z_1=0$ and $z_2=6+8i$.
Find the midpoint of $z_1=0$ and $z_2=6+8i$.
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$3+4i$. $\frac{0+(6+8i)}{2} = \frac{6+8i}{2} = 3+4i$.
$3+4i$. $\frac{0+(6+8i)}{2} = \frac{6+8i}{2} = 3+4i$.
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Find the distance between $z_1=0+2i$ and $z_2=0-4i$.
Find the distance between $z_1=0+2i$ and $z_2=0-4i$.
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$6$. $|(0+2i)-(0-4i)| = |6i| = 6$.
$6$. $|(0+2i)-(0-4i)| = |6i| = 6$.
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Find the midpoint of $z_1=-7+3i$ and $z_2=1-5i$.
Find the midpoint of $z_1=-7+3i$ and $z_2=1-5i$.
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$-3-i$. $\frac{(-7+3i)+(1-5i)}{2} = \frac{-6-2i}{2} = -3-i$.
$-3-i$. $\frac{(-7+3i)+(1-5i)}{2} = \frac{-6-2i}{2} = -3-i$.
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What is the distance between $z_1$ and $z_2$ if $z_1-z_2=-8+6i$?
What is the distance between $z_1$ and $z_2$ if $z_1-z_2=-8+6i$?
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$10$. $|-8+6i| = \sqrt{64+36} = 10$.
$10$. $|-8+6i| = \sqrt{64+36} = 10$.
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Find the distance between $z_1=2-4i$ and $z_2=-6+0i$.
Find the distance between $z_1=2-4i$ and $z_2=-6+0i$.
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$\sqrt{80}$. $|(2-4i)-(-6+0i)| = |8-4i| = \sqrt{64+16} = \sqrt{80}$.
$\sqrt{80}$. $|(2-4i)-(-6+0i)| = |8-4i| = \sqrt{64+16} = \sqrt{80}$.
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Find the distance between $z_1=-3+0i$ and $z_2=5+0i$.
Find the distance between $z_1=-3+0i$ and $z_2=5+0i$.
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$8$. $|(-3+0i)-(5+0i)| = |-8| = 8$.
$8$. $|(-3+0i)-(5+0i)| = |-8| = 8$.
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Find $|z|$ for $z=9+12i$.
Find $|z|$ for $z=9+12i$.
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$15$. $|9+12i| = \sqrt{81+144} = \sqrt{225} = 15$.
$15$. $|9+12i| = \sqrt{81+144} = \sqrt{225} = 15$.
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Find $|z|$ for $z=5-12i$.
Find $|z|$ for $z=5-12i$.
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$13$. $|5-12i| = \sqrt{25+144} = \sqrt{169} = 13$.
$13$. $|5-12i| = \sqrt{25+144} = \sqrt{169} = 13$.
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Find the distance between $z_1=1+2i$ and $z_2=4+6i$.
Find the distance between $z_1=1+2i$ and $z_2=4+6i$.
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$5$. $|(1+2i)-(4+6i)| = |-3-4i| = \sqrt{9+16} = 5$.
$5$. $|(1+2i)-(4+6i)| = |-3-4i| = \sqrt{9+16} = 5$.
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Find the distance between $z_1=-2+1i$ and $z_2=4-7i$.
Find the distance between $z_1=-2+1i$ and $z_2=4-7i$.
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$10$. $|(-2+1i)-(4-7i)| = |-6+8i| = \sqrt{36+64} = 10$.
$10$. $|(-2+1i)-(4-7i)| = |-6+8i| = \sqrt{36+64} = 10$.
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Find the midpoint of $z_1=1+2i$ and $z_2=9+10i$.
Find the midpoint of $z_1=1+2i$ and $z_2=9+10i$.
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$5+6i$. $\frac{(1+2i)+(9+10i)}{2} = \frac{10+12i}{2} = 5+6i$.
$5+6i$. $\frac{(1+2i)+(9+10i)}{2} = \frac{10+12i}{2} = 5+6i$.
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What is the distance between $z_1$ and $z_2$ if $z_1-z_2=3-4i$?
What is the distance between $z_1$ and $z_2$ if $z_1-z_2=3-4i$?
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$5$. $|3-4i| = \sqrt{9+16} = 5$.
$5$. $|3-4i| = \sqrt{9+16} = 5$.
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Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=0+7i$.
Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=0+7i$.
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$7$. $|0+7i| = |7i| = 7$.
$7$. $|0+7i| = |7i| = 7$.
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Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=-9+0i$.
Identify the distance between $z_1$ and $z_2$ if $z_1-z_2=-9+0i$.
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$9$. $|-9+0i| = |-9| = 9$.
$9$. $|-9+0i| = |-9| = 9$.
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Find the midpoint of $z_1=2-4i$ and $z_2=-6+0i$.
Find the midpoint of $z_1=2-4i$ and $z_2=-6+0i$.
Tap to reveal answer
$-2-2i$. $\frac{(2-4i)+(-6+0i)}{2} = \frac{-4-4i}{2} = -2-2i$.
$-2-2i$. $\frac{(2-4i)+(-6+0i)}{2} = \frac{-4-4i}{2} = -2-2i$.
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Find the distance between $z_1=2-4i$ and $z_2=-6+0i$.
Find the distance between $z_1=2-4i$ and $z_2=-6+0i$.
Tap to reveal answer
$\sqrt{80}$. $|(2-4i)-(-6+0i)| = |8-4i| = \sqrt{64+16} = \sqrt{80}$.
$\sqrt{80}$. $|(2-4i)-(-6+0i)| = |8-4i| = \sqrt{64+16} = \sqrt{80}$.
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Find the distance between $z_1=3+0i$ and $z_2=0+4i$.
Find the distance between $z_1=3+0i$ and $z_2=0+4i$.
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$5$. $|(3+0i)-(0+4i)| = |3-4i| = \sqrt{9+16} = 5$.
$5$. $|(3+0i)-(0+4i)| = |3-4i| = \sqrt{9+16} = 5$.
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Find the midpoint of $z_1=3+0i$ and $z_2=0+4i$.
Find the midpoint of $z_1=3+0i$ and $z_2=0+4i$.
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$\frac{3}{2}+2i$. $\frac{(3+0i)+(0+4i)}{2} = \frac{3+4i}{2} = \frac{3}{2}+2i$.
$\frac{3}{2}+2i$. $\frac{(3+0i)+(0+4i)}{2} = \frac{3+4i}{2} = \frac{3}{2}+2i$.
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Identify the real part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.
Identify the real part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.
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$\frac{a+c}{2}$. Average the real parts: $\frac{a+c}{2}$.
$\frac{a+c}{2}$. Average the real parts: $\frac{a+c}{2}$.
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Identify the imaginary part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.
Identify the imaginary part of the midpoint $m=\frac{z_1+z_2}{2}$ for $z_1=a+bi$ and $z_2=c+di$.
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$\frac{b+d}{2}$. Average the imaginary parts: $\frac{b+d}{2}$.
$\frac{b+d}{2}$. Average the imaginary parts: $\frac{b+d}{2}$.
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What is the midpoint of $z_1=a$ and $z_2=b$ when both endpoints are real numbers?
What is the midpoint of $z_1=a$ and $z_2=b$ when both endpoints are real numbers?
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$\frac{a+b}{2}$. For real numbers, the midpoint formula reduces to the average.
$\frac{a+b}{2}$. For real numbers, the midpoint formula reduces to the average.
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What is the distance between $z_1=a$ and $z_2=b$ when both endpoints are real numbers?
What is the distance between $z_1=a$ and $z_2=b$ when both endpoints are real numbers?
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$|a-b|$. For real numbers, distance is the absolute value of the difference.
$|a-b|$. For real numbers, distance is the absolute value of the difference.
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Find the distance between $z_1=5$ and $z_2=-1+4i$.
Find the distance between $z_1=5$ and $z_2=-1+4i$.
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$\sqrt{52}$. $|5-(-1+4i)| = |6-4i| = \sqrt{36+16} = \sqrt{52}$.
$\sqrt{52}$. $|5-(-1+4i)| = |6-4i| = \sqrt{36+16} = \sqrt{52}$.
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Find the midpoint of $z_1=-3i$ and $z_2=9i$.
Find the midpoint of $z_1=-3i$ and $z_2=9i$.
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$3i$. $\frac{-3i+9i}{2} = \frac{6i}{2} = 3i$.
$3i$. $\frac{-3i+9i}{2} = \frac{6i}{2} = 3i$.
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Find the midpoint of $z_1=5$ and $z_2=-1+4i$.
Find the midpoint of $z_1=5$ and $z_2=-1+4i$.
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$2+2i$. $\frac{5+(-1+4i)}{2} = \frac{4+4i}{2} = 2+2i$.
$2+2i$. $\frac{5+(-1+4i)}{2} = \frac{4+4i}{2} = 2+2i$.
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State the formula for the distance between complex numbers $z_1$ and $z_2$ in the complex plane.
State the formula for the distance between complex numbers $z_1$ and $z_2$ in the complex plane.
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$d=|z_1-z_2|$. Distance is the modulus of the difference between two complex numbers.
$d=|z_1-z_2|$. Distance is the modulus of the difference between two complex numbers.
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State the formula for the midpoint of endpoints $z_1$ and $z_2$ in the complex plane.
State the formula for the midpoint of endpoints $z_1$ and $z_2$ in the complex plane.
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$m=\frac{z_1+z_2}{2}$. Average the two endpoints by adding and dividing by 2.
$m=\frac{z_1+z_2}{2}$. Average the two endpoints by adding and dividing by 2.
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What is the modulus of $z=a+bi$ written in terms of $a$ and $b$?
What is the modulus of $z=a+bi$ written in terms of $a$ and $b$?
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$|z|=\sqrt{a^2+b^2}$. Modulus uses the Pythagorean theorem with real and imaginary parts.
$|z|=\sqrt{a^2+b^2}$. Modulus uses the Pythagorean theorem with real and imaginary parts.
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What is the distance from the origin to $z$ in the complex plane, written using modulus?
What is the distance from the origin to $z$ in the complex plane, written using modulus?
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$|z|$. The modulus gives the distance from any point to the origin.
$|z|$. The modulus gives the distance from any point to the origin.
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