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MATH / Let z=2+√(2)−(3+3√(2))i, and let c=2−3i. Let w be the result when z is rotated around c by (π)/(4) counter-clockwise. [asy] unitsize(0.6 cm); pair C, W, Z; Z = (2 + sqrt(2), -3 - 3…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let z=2+2−(3+32)iz = 2 + \sqrt{2} - (3 + 3 \sqrt{2})i, and let c=2−3ic = 2 - 3i. Let ww be the result when zz is rotated around cc by π4\frac{\pi}{4} counter-clockwise. [asy] unitsize(0.6 cm); pair C, W, Z; Z = (2 + sqrt(2), -3 - 3*sqrt(2)); C = (2,-3); W = rotate(45,C)*(Z); draw(Z--C--W); dot("cc", C, N); dot("ww", W, SE); dot("zz", Z, S); label("π4\frac{\pi}{4}", C + (0.6,-1)); [/asy] Find w.w.
Plain-text mathematical notation (without MathML)
Let z=2+√(2)−(3+3√(2))i, and let c=2−3i.  Let w be the result when z is rotated around c by (π)/(4) counter-clockwise.

[asy]
unitsize(0.6 cm);

pair C, W, Z;

Z = (2 + sqrt(2), -3 - 3*sqrt(2));
C = (2,-3);
W = rotate(45,C)*(Z);

draw(Z--C--W);

dot("c", C, N);
dot("w", W, SE);
dot("z", Z, S);
label("(π)/(4)", C + (0.6,-1));
[/asy]

Find w.
Original LaTeX notation
Let $z = 2 + \sqrt{2} - (3 + 3 \sqrt{2})i$, and let $c = 2 - 3i$.  Let $w$ be the result when $z$ is rotated around $c$ by $\frac{\pi}{4}$ counter-clockwise.

[asy]
unitsize(0.6 cm);

pair C, W, Z;

Z = (2 + sqrt(2), -3 - 3*sqrt(2));
C = (2,-3);
W = rotate(45,C)*(Z);

draw(Z--C--W);

dot("$c$", C, N);
dot("$w$", W, SE);
dot("$z$", Z, S);
label("$\frac{\pi}{4}$", C + (0.6,-1));
[/asy]

Find $w.$

level

Level 5

type

Precalculus

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Source and history

task_id
fa2ee8c2-c607-5af8-953e-839cfb500e2d
task_key
precalculus--test--fa2ee8c2-c607-5af8-953e-839cfb500e2d
task_revision_id
1
config / split
precalculus / test

Official source

initial import