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Difference between revisions of "2000 AMC 12 Problems/Problem 10"

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==Solution==
 
==Solution==
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Step 1: Reflect in the xy plane. That is the same as creating the additive inverse of z and sticking it into the z coordinate. <math>(1,2,-3)</math>
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Step 1: Reflect in the <math>xy</math>-plane. Replace <math>z</math> with its additive inverse: <math>(1,2,-3)</math>
  
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Step 2: Rotate around x-axis 180 degrees. That is the same as taking the additive inverse of y and sticking it in the y coordinate, then taking the additive inverse of z and sticking it into the z coordinate. <math>(1, -2, 3)</math>
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Step 2: Rotate around <math>x</math>-axis 180 degrees. Replace <math>y</math> and <math>z</math> with their respective additive inverses. <math>(1, -2, 3)</math>
  
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Step 3: Translate 5 units in positive-y direction. That is the same as adding 5 to y and sticking it in the y coordinate. <math>(1,3,3) \Rightarrow \text {(E) }</math>
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Step 3: Translate <math>5</math> units in positive-<math>y</math> direction. Replace <math>y</math> with <math>y+5</math>. <math>(1,3,3) \Rightarrow \text {(E) }</math>
  
 
==See Also==
 
==See Also==
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[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
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[[Category:3D Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 08:56, 21 August 2023

Problem

The point $P = (1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by 5 units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$?

$\text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1, - 2,8) \qquad \text {(D) } ( - 1,3,3) \qquad \text {(E) } (1,3,3)$

Solution

Step 1: Reflect in the $xy$-plane. Replace $z$ with its additive inverse: $(1,2,-3)$

Step 2: Rotate around $x$-axis 180 degrees. Replace $y$ and $z$ with their respective additive inverses. $(1, -2, 3)$

Step 3: Translate $5$ units in positive-$y$ direction. Replace $y$ with $y+5$. $(1,3,3) \Rightarrow \text {(E) }$

See Also

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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