2021 AMC 12B Problems/Problem 5: Difference between revisions

Latest revision as of 17:28, 19 October 2025

The following problem is from both the 2021 AMC 10B #9 and 2021 AMC 12B #5, so both problems redirect to this page.

Problem

The point $P(a,b)$ in the $xy$ -plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$ . The image of $P$ after these two transformations is at $(-6,3)$ . What is $b - a ?$

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~5 \qquad\textbf{(D)} ~7 \qquad\textbf{(E)} ~9$

Solution 1 (Transformation Rules)

The final image of $P$ is $(-6,3)$ . We know the reflection rule for reflecting over $y=-x$ is $(x,y) \rightarrow (-y, -x)$ . So before the reflection and after rotation the point is $(-3,6)$ .

By definition of rotation, the slope between $(-3,6)$ and $(1,5)$ must be perpendicular to the slope between $(a,b)$ and $(1,5)$ . The first slope is $\frac{5-6}{1-(-3)} = \frac{-1}{4}$ . This means the slope of $P$ and $(1,5)$ is $4$ .

Rotations also preserve distance to the center of rotation, and since we only "travelled" up and down by the slope once to get from $(-3,6)$ to $(1,5)$ it follows we shall only use the slope once to travel from $(1,5)$ to $P$ .

Therefore point $P$ is located at $(1+1, 5+4) = (2,9)$ . The answer is $9-2 = 7 = \boxed{\textbf{(D)} ~7}$ .

-abhinavg0627

Solution 2 (Complex Numbers)

Let us reconstruct that coordinate plane as the complex plane. Then, the point $P(a, b)$ becomes $a+b\cdot{i}$ . A $90^\circ$ rotation around the point $(1, 5)$ can be done by translating the point $(1, 5)$ to the origin, rotating around the origin by $90^\circ$ , and then translating the origin back to the point $(1, 5)$ . $a+b\cdot{i} \implies (a-1)+(b-5)\cdot{i} \implies ((a-1)+(b-5)\cdot{i})\cdot{i} = 5-b+(a-1)i \implies 5+1-b+(a-1+5)i = 6-b+(a+4)i.$ By basis reflection rules, the reflection of $(-6, 3)$ about the line $y = -x$ is $(-3, 6)$ . Hence, we have $6-b+(a+4)i = -3+6i \implies b=9, a=2,$ from which $b-a = 9-2 = \boxed{\textbf{(D)} ~7}$ .

~twotothetenthis1024

Solution 3 (Reverso)

The problem gives a series of transformations and proceeds to give the resultant point, being $(-6,3)=P$ . Therefore, all we must do is reverse it. First, we reverse the last transformation by computing the distance from the point $(-6,3)$ to $y=-x$ by using the formula $d=\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}$ . Where $Ax+By+C$ is the standard form of a line. Computing, we obtain that $d=\frac{3\sqrt{2}}{2}$ . We also know that this is magnitude is at an angle of $\frac{\pi}{4}$ . Therefore, to do the transformation, we double that vector and add it to the point. We get:

$P^{'}=P+2\overrightarrow{v}$ $\implies P'=(-6,3)+\left\langle 3\sqrt{2}\cdot \frac{\sqrt{2}}{2},3\sqrt{2}\cdot \frac{\sqrt{2}}{2} \right \rangle$ $\implies P'=(-3,6)$ .

Now, we must reverse the second transformation. To do so, realize that $P'-(1,5)\Leftrightarrow \overrightarrow{v_{2}}=\left\langle -4,1 \right\rangle$ . Simply make this vector perpendicular by switching the $x$ and $y$ components and switching the sign of the initial $y$ component. Therefore, we get $\overrightarrow{v_{2\bot }}=\left\langle 1,4 \right\rangle$ . Therefore, adding this vector to $(1,5)$ yields $(2,9)$ , which leads us to $9-2 = \boxed{\textbf{(D)} ~7}$ .

~justgiveup JoshKosh

Solution 4 (Vector Dot Product)

Using the same method as in Solution 1, we can obtain that the point before the reflection is $(-3,6)$ . If we let the original point be $(x, y)$ , then we can use that the starting point is $(1,5)$ to obtain two vectors $\langle -4,1 \rangle$ and $\langle x-1, y-5 \rangle$ . We know that two vectors are perpendicular if their dot product is equal to $0$ , and that both points are the same distance ( $\sqrt {17}$ ) from $(1,5)$ .

Therefore, we can write two equations using these vectors: $(x-1)^2 + (y-5)^2 = 17$ (from distance and pythagorean theorem) and $-4x+y-1 = 0$ (from dot product)

Solving, we simplify the second equation to $y=4x+1$ , and plug it into the first equation. We obtain $(x-1)^2 + (4x-4)^2 = 17$ . We can simplify this to the quadratic $17x^2-34x=0$ . When we factor out $17x$ , we find that $x = 2$ or $x = 0$ . However, $x$ cannot equal $0$ . Therefore, $x = 2$ , and plugging this into the second equation gives us that $y = 9$ . Since the point is $(9, 2)$ , we compute $9-2 = \boxed{\textbf{(D)} ~7}$ .

~saturnrocket

Solution 5 (Vector Dot Product scuffed version)

Using the same method as in Solution 1 reflecting $(-6,3)$ about the line $y = -x$ gives us $(-3,6).$

Let the original point be $\langle x,y \rangle.$ From point $(1,5),$ we form the vectors $\langle -4,1 \rangle$ and $\langle x-1, y-5 \rangle$ that extend out from the initial point. If they are perpendicular, we know that their dot product has to equal zero. Therefore, $\langle -4,1 \rangle \cdot \langle x-1, y-5 \rangle = 0 \implies -4x+y-1= 0.$ Now, we have to do some guess and check from the multiple choices. Let $y - x = A$ where $A$ is one of the answer choices. Then, $A -3x = 1.$ By intuition and logical reasoning we deduce that $A$ must be $1 \pmod 3$ so that brings our potential answers down to $\text{\textbf{(A)}}$ and $\text{\textbf{(C)}}.$ If $A = 1$ from $\text{\textbf{(A)}},$ then $x = 0,$ which we can quickly rule out since we know thar $P$ rotated counterclockwise not clockwise. Hence, $\boxed{\textbf{(D)} ~7}$ is the answer.

~peelybonehead

Solution 6 (Working Backwards Approach, Super Easy!)

The last thing that happened was the reflection over the line $y = -x$ , so we reverse that by reflecting $(-6,3)$ over $y = -x$ to get $(-3,6)$ . Now, we have to rotate it $\textbf{clockwise}$ , to reverse the first step. To rotate $(-3,6)$ over the point $(1,5)$ clockwise 90 degrees, we can subtract $1$ from each x-coordinate and subtract $5$ from each y-coordinate. That way, we only have to rotate $(-4,1)$ clockwise 90 degrees over the origin $(0,0)$ . This is pretty easy to do in your head, so you get $(1,4)$ . But we are not, done, we still have to add $1$ to each x-coordinate and add $5$ to each y-coordinate, to get $(2,9)$ , so our final answer is $9 - 2 = 7$ , which is $\boxed{\textbf{(D)} ~7}$ .

~NXC

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=qpvS2PVkI8A&t=335s

Video Solution by OmegaLearn (Rotation & Reflection tricks)

https://youtu.be/VyRWjgGIsRQ

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=VzwxbsuSQ80

Video Solution by TheBeautyofMath

https://youtu.be/GYpAm8v1h-U?t=860 (for AMC 10B)

https://youtu.be/EMzdnr1nZcE?t=814 (for AMC 12B)

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=776

~Interstigation

Video Solution (Just 3 min!)

https://youtu.be/j39KCUC2Qz8

~Education, the Study of Everything

@@ Line 1: / Line 1: @@
-The 2021 AMC 12B will be held on February 10th, 2021. The problems will not be made public until 24 hours after that.
+{{duplicate|[[2021 AMC 10B Problems#Problem 9|2021 AMC 10B #9]] and [[2021 AMC 12B Problems#Problem 5|2021 AMC 12B #5]]}}
+==Problem==
+The point <math>P(a,b)</math> in the <math>xy</math>-plane is first rotated counterclockwise by <math>90^\circ</math> around the point <math>(1,5)</math> and then reflected about the line <math>y = -x</math>. The image of <math>P</math> after these two transformations is at <math>(-6,3)</math>. What is <math>b - a ?</math>
+<math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~5 \qquad\textbf{(D)} ~7 \qquad\textbf{(E)} ~9</math>
+==Solution 1 (Transformation Rules)==
+The final image of <math>P</math> is <math>(-6,3)</math>. We know the reflection rule for reflecting over <math>y=-x</math> is <math>(x,y) \rightarrow (-y, -x)</math>. So before the reflection and after rotation the point is <math>(-3,6)</math>.
+By definition of rotation, the slope between <math>(-3,6)</math> and <math>(1,5)</math> must be perpendicular to the slope between <math>(a,b)</math> and <math>(1,5)</math>. The first slope is <math>\frac{5-6}{1-(-3)} = \frac{-1}{4}</math>. This means the slope of <math>P</math> and <math>(1,5)</math> is <math>4</math>.
+Rotations also preserve distance to the center of rotation, and since we only "travelled" up and down by the slope once to get from <math>(-3,6)</math> to <math>(1,5)</math> it follows we shall only use the slope once to travel from <math>(1,5)</math> to <math>P</math>.
+Therefore point <math>P</math> is located at <math>(1+1, 5+4) = (2,9)</math>. The answer is <math>9-2 = 7 = \boxed{\textbf{(D)} ~7}</math>.
+-abhinavg0627
+==Solution 2 (Complex Numbers)==
+Let us reconstruct that coordinate plane as the complex plane. Then, the point <math>P(a, b)</math> becomes <math>a+b\cdot{i}</math>.
+A <math>90^\circ</math> rotation around the point <math>(1, 5)</math> can be done by translating the point <math>(1, 5)</math> to the origin, rotating around the origin by
+<math>90^\circ</math>, and then translating the origin back to the point <math>(1, 5)</math>.
+<cmath>a+b\cdot{i}  \implies (a-1)+(b-5)\cdot{i} \implies ((a-1)+(b-5)\cdot{i})\cdot{i} = 5-b+(a-1)i \implies 5+1-b+(a-1+5)i = 6-b+(a+4)i.</cmath>
+By basis reflection rules, the reflection of <math>(-6, 3)</math> about the line <math>y = -x</math> is <math>(-3, 6)</math>.
+Hence, we have <cmath>6-b+(a+4)i = -3+6i \implies b=9, a=2,</cmath> from which <math>b-a = 9-2 = \boxed{\textbf{(D)} ~7}</math>.
+~twotothetenthis1024
+==Solution 3 (Reverso)==
+The problem gives a series of transformations and proceeds to give the resultant point, being <math>(-6,3)=P</math>. Therefore, all we must do is reverse it. First, we reverse the last transformation by computing the distance from the point <math>(-6,3)</math> to <math>y=-x</math> by using the formula <math>d=\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}</math>. Where <math>Ax+By+C</math> is the standard form of a line. Computing, we obtain that <math>d=\frac{3\sqrt{2}}{2}</math>. We also know that this is magnitude is at an angle of <math>\frac{\pi}{4}</math>. Therefore, to do the transformation, we double that vector and add it to the point. We get:
+<math>P^{'}=P+2\overrightarrow{v}</math>
+<math>\implies P'=(-6,3)+\left\langle 3\sqrt{2}\cdot \frac{\sqrt{2}}{2},3\sqrt{2}\cdot \frac{\sqrt{2}}{2} \right \rangle </math>
+<math>\implies P'=(-3,6)</math>.
+Now, we must reverse the second transformation. To do so, realize that <math>P'-(1,5)\Leftrightarrow \overrightarrow{v_{2}}=\left\langle -4,1 \right\rangle </math>. Simply make this vector perpendicular by switching the <math>x</math> and <math>y</math> components and switching the sign of the initial <math>y</math> component. Therefore, we get <math>\overrightarrow{v_{2\bot }}=\left\langle 1,4 \right\rangle </math>. Therefore, adding this vector to <math>(1,5)</math> yields <math>(2,9)</math>, which leads us to <math>9-2 = \boxed{\textbf{(D)} ~7}</math>.
+~justgiveup JoshKosh
+==Solution 4 (Vector Dot Product)==
+Using the same method as in Solution 1, we can obtain that the point before the reflection is <math>(-3,6)</math>.
+If we let the original point be <math>(x, y)</math>, then we can use that the starting point is <math>(1,5)</math> to obtain two vectors <math>\langle -4,1 \rangle</math> and <math>\langle x-1, y-5 \rangle</math>.
+We know that two vectors are perpendicular if their dot product is equal to <math>0</math>, and that both points are the same distance (<math>   \sqrt {17}</math>) from <math>(1,5)</math>.
+Therefore, we can write two equations using these vectors:
+<math>(x-1)^2 + (y-5)^2 = 17</math> (from distance and pythagorean theorem) and <math>-4x+y-1 = 0</math> (from dot product)
+Solving, we simplify the second equation to <math>y=4x+1</math>, and plug it into the first equation.
+We obtain <math>(x-1)^2 + (4x-4)^2 = 17</math>. We can simplify this to the quadratic <math>17x^2-34x=0</math>.
+When we factor out <math>17x</math>, we find that <math>x = 2</math> or <math>x = 0</math>. However, <math>x</math> cannot equal <math>0</math>.
+Therefore, <math>x = 2</math>, and plugging this into the second equation gives us that <math>y = 9</math>.
+Since the point is <math>(9, 2)</math>, we compute <math>9-2 = \boxed{\textbf{(D)} ~7}</math>.
+~saturnrocket
+==Solution 5 (Vector Dot Product scuffed version)==
+Using the same method as in Solution 1 reflecting <math>(-6,3)</math> about the line <math>y = -x</math> gives us <math>(-3,6).</math>
+Let the original point be <math>\langle x,y \rangle.</math> From point <math>(1,5),</math> we form the vectors <math>\langle -4,1 \rangle</math> and <math>\langle x-1, y-5 \rangle</math> that extend out from the initial point. If they are perpendicular, we know that their dot product has to equal zero. Therefore, <cmath>\langle -4,1 \rangle \cdot \langle x-1, y-5 \rangle = 0 \implies -4x+y-1= 0.</cmath>Now, we have to do some guess and check from the multiple choices. Let <math>y - x = A</math> where <math>A</math> is one of the answer choices. Then, <math>A -3x = 1.</math> By intuition and logical reasoning we deduce that <math>A</math> must be <math>1 \pmod 3</math> so that brings our potential answers down to <math>\text{\textbf{(A)}}</math> and <math>\text{\textbf{(C)}}.</math> If <math>A = 1</math> from <math>\text{\textbf{(A)}},</math> then <math>x = 0,</math> which we can quickly rule out since we know thar <math>P</math> rotated counterclockwise not clockwise. Hence, <math>\boxed{\textbf{(D)} ~7}</math> is the answer.
+~peelybonehead
+== Solution 6 (Working Backwards Approach, Super Easy!) ==
+The last thing that happened was the reflection over the line <math>y = -x</math>, so we reverse that by reflecting <math>(-6,3)</math> over <math>y = -x</math> to get <math>(-3,6)</math>. Now, we have to rotate it <math>\textbf{clockwise}</math>, to reverse the first step. To rotate <math>(-3,6)</math> over the point <math>(1,5)</math> clockwise 90 degrees, we can subtract <math>1</math> from each x-coordinate and subtract <math>5</math> from each y-coordinate. That way, we only have to rotate <math>(-4,1)</math> clockwise 90 degrees over the origin <math>(0,0)</math>. This is pretty easy to do in your head, so you get <math>(1,4)</math>. But we are not, done, we still have to add <math>1</math> to each x-coordinate and add <math>5</math> to each y-coordinate, to get <math>(2,9)</math>, so our final answer is <math>9 - 2 = 7</math>, which is <math>\boxed{\textbf{(D)} ~7}</math>.
+~NXC
+==Video Solution by Punxsutawney Phil==
+https://youtube.com/watch?v=qpvS2PVkI8A&t=335s
+== Video Solution by OmegaLearn (Rotation & Reflection tricks) ==
+https://youtu.be/VyRWjgGIsRQ
+~ pi_is_3.14
+==Video Solution by Hawk Math==
+https://www.youtube.com/watch?v=VzwxbsuSQ80
+==Video Solution by TheBeautyofMath==
+https://youtu.be/GYpAm8v1h-U?t=860 (for AMC 10B)
+https://youtu.be/EMzdnr1nZcE?t=814 (for AMC 12B)
+~IceMatrix
+==Video Solution by Interstigation==
+https://youtu.be/DvpN56Ob6Zw?t=776
+~Interstigation
+==Video Solution (Just 3 min!)==
+https://youtu.be/j39KCUC2Qz8
+~Education, the Study of Everything
+==See Also==
+{{AMC12 box|year=2021|ab=B|num-b=4|num-a=6}}
+{{AMC10 box|year=2021|ab=B|num-b=8|num-a=10}}
+{{MAA Notice}}

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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All AMC 12 Problems and Solutions

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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 10 Problems and Solutions

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