AMC 10 · 2024 · #13

Grade 8 geometry-2dcounting

Archetype Small-Domain Constrained Search

coordinate-geometryreflection-symmetrysystematic-enumerationcombinations-basic caseworksystematic-enumeration ↑ Prerequisites: coordinate-geometryreflection-symmetry

📏 Long solution 💡 3 insights

Problem

Two transformations are said to commute if applying the first followed by the second
gives the same result as applying the second followed by the first. Consider these
four transformations of the coordinate plane:

a translation $2$ units to the right,
a $90^{\circ}$ -rotation counterclockwise about the origin,
a reflection across the $x$ -axis, and
a dilation centered at the origin with scale factor $2.$

Of the $6$ pairs of distinct transformations from this list, how many commute?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Four plane transformations are given: $T_1$ = translate $2$ units right, $T_2$ = rotate $90^{\circ}$ counterclockwise about the origin, $T_3$ = reflect across the $x$-axis, $T_4$ = dilate from the origin by factor $2$. Two transformations commute if applying them in either order gives the same result. Of the $\binom{4}{2} = 6$ pairs, how many commute?

Givens: $T_1(x, y) = (x+2, y)$ — translate right by $2$; $T_2(x, y) = (-y, x)$ — rotate $90^{\circ}$ counterclockwise about origin; $T_3(x, y) = (x, -y)$ — reflect across $x$-axis; $T_4(x, y) = (2x, 2y)$ — dilate from origin by factor $2$; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$

Unknowns: The number of pairs $(T_i, T_j)$ with $i < j$ for which $T_i \circ T_j = T_j \circ T_i$ on every point $(x,y)$

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #13 Convert to Algebra

The question itself names a finite collection — $\binom{4}{2} = 6$ pairs — and asks how many satisfy a yes/no condition. That is the signature of Tool #2 (Make a Systematic List): enumerate every pair, decide each one, then count the yeses. The deciding mechanism is Tool #13 (Convert to Algebra): once each geometric move is written as a coordinate rule, "do they commute" reduces to comparing two ordered pairs symbolically. Algebra is the right tool here because tracing pictures by hand for six pairs would be slow and error-prone, while the coordinate rules let each check fit on one line.

Execute — Answer: C

#13 Convert to Algebra 8.G.A.3 Step 1

Write each move as a coordinate rule.
Reading off the geometric description: translating right adds $2$ to the $x$-coordinate; rotating $90^{\circ}$ counterclockwise sends $(x,y)$ to $(-y, x)$; reflecting across the $x$-axis flips the sign of $y$; dilating by factor $2$ from the origin doubles both coordinates.

$$T_1(x,y) = (x+2, y), \;\; T_2(x,y) = (-y, x), \;\; T_3(x,y) = (x, -y), \;\; T_4(x,y) = (2x, 2y)$$

💡 Grade 8 transformation rules: each move becomes a formula on coordinates, so composing two moves is just substitution.

#2 Make a Systematic List 7.SP.C.8 Step 2

List the $6$ pairs in order so none is missed.
With four moves $T_1, T_2, T_3, T_4$, the pairs (smaller index first) are $(T_1,T_2), (T_1,T_3), (T_1,T_4), (T_2,T_3), (T_2,T_4), (T_3,T_4)$.
For each pair we will compute both orders of composition on $(x,y)$ and compare.
A pair commutes iff the two results match for every $(x,y)$.

$$\text{Pairs to test: } \{1{,}2\}, \{1{,}3\}, \{1{,}4\}, \{2{,}3\}, \{2{,}4\}, \{3{,}4\}$$

💡 Listing pairs in index order is the simple-event-tree habit from Grade 7 probability: every pair appears exactly once.

#13 Convert to Algebra 6.EE.A.4 Step 3

Pair $\{1,2\}$ — translate vs.
rotate.
Compose both ways and compare.
The two outputs differ in both coordinates, so this pair does not commute.
Geometric reason: translating then rotating moves the translation vector along with the rotation, while rotating then translating leaves the original $2$-unit offset on the rotated point — different.

$T_2(T_1(x,y)) = T_2(x+2, y) = (-y, x+2)$\\$T_1(T_2(x,y)) = T_1(-y, x) = (-y+2, x)$\\$(-y, x+2) \neq (-y+2, x) \;\Rightarrow\; \text{not commute}$

💡 Grade 6 "are these two expressions equivalent for all values" check — a single counterexample value of $(x,y)$ would disprove it; here the expressions themselves differ.

#13 Convert to Algebra 6.EE.A.4 Step 4

Pair $\{1,3\}$ — translate vs.
reflect across $x$-axis.
The translation only affects $x$ and the reflection only affects $y$, so they touch disjoint coordinates and the order cannot matter.
The algebra confirms it.

$T_3(T_1(x,y)) = T_3(x+2, y) = (x+2, -y)$\\$T_1(T_3(x,y)) = T_1(x, -y) = (x+2, -y)$\\$\text{Equal} \;\Rightarrow\; \text{commute}$

💡 Operations on independent coordinates always commute — like first putting on a left sock and then a right shoe vs. the reverse.

#13 Convert to Algebra 8.G.A.3 Step 5

Pair $\{1,4\}$ — translate vs.
dilate.
Dilation from the origin scales the translation vector too, but pure translation does not, so the order matters.

$T_4(T_1(x,y)) = T_4(x+2, y) = (2x+4, 2y)$\\$T_1(T_4(x,y)) = T_1(2x, 2y) = (2x+2, 2y)$\\$(2x+4, 2y) \neq (2x+2, 2y) \;\Rightarrow\; \text{not commute}$

💡 Dilation from the origin treats the $2$-unit shift like every other length and doubles it; translation does not get a second chance to double itself.

#13 Convert to Algebra 8.G.A.3 Step 6

Pair $\{2,3\}$ — rotate $90^{\circ}$ CCW vs.
reflect across $x$-axis.
Both touch both coordinates, so we cannot wave the result away — we compute.

$T_3(T_2(x,y)) = T_3(-y, x) = (-y, -x)$\\$T_2(T_3(x,y)) = T_2(x, -y) = (-(-y), x) = (y, x)$\\$(-y, -x) \neq (y, x) \;\Rightarrow\; \text{not commute}$

💡 Geometrically, rotate-then-reflect and reflect-then-rotate give two different reflections (across perpendicular lines), so the outputs disagree.

#13 Convert to Algebra 8.G.A.3 Step 7

Pair $\{2,4\}$ — rotate $90^{\circ}$ CCW vs.
dilate from origin.
Both fix the origin and the rotation only swaps and negates coordinates while dilation scales them uniformly, so the order should not matter.
Algebra confirms it.

$T_4(T_2(x,y)) = T_4(-y, x) = (-2y, 2x)$\\$T_2(T_4(x,y)) = T_2(2x, 2y) = (-2y, 2x)$\\$\text{Equal} \;\Rightarrow\; \text{commute}$

💡 Scaling by a constant and applying a linear coordinate swap can be done in either order — multiplication by $2$ slides through the rotation.

#13 Convert to Algebra 6.EE.A.3 Step 8

Pair $\{3,4\}$ — reflect across $x$-axis vs.
dilate from origin.
Reflection multiplies $y$ by $-1$ and dilation multiplies both coordinates by $2$.
Since multiplication by $-1$ and by $2$ commute, the combined order does too.

$T_4(T_3(x,y)) = T_4(x, -y) = (2x, -2y)$\\$T_3(T_4(x,y)) = T_3(2x, 2y) = (2x, -2y)$\\$\text{Equal} \;\Rightarrow\; \text{commute}$

💡 Coordinate-wise the operations are just $\times(-1)$ on $y$ and $\times 2$ on both — Grade 6 properties of multiplication say constant factors can be reordered.

#2 Make a Systematic List 4.OA.C.5 Step 9

Count the commuting pairs from the list.
From the six rows above, the ones marked "commute" are $\{1,3\}, \{2,4\}, \{3,4\}$.
That is $3$ pairs out of $6$.

$$\text{Commuting pairs} = \{\{1,3\}, \{2,4\}, \{3,4\}\}, \;\; \text{count} = 3 \;\Rightarrow\; \textbf{(C)}$$

💡 After listing each case as commute / not commute, the final count is a Grade 4 "how many in the list satisfy the rule" tally.

[1] #13 8.G.A.3 Write each move as a coordinate rule. Reading off the geometric description: tra

[2] #2 7.SP.C.8 List the $6$ pairs in order so none is missed. With four moves $T_1, T_2, T_3, T

[3] #13 6.EE.A.4 Pair $\{1,2\}$ — translate vs. rotate. Compose both ways and compare. The two ou

[4] #13 6.EE.A.4 Pair $\{1,3\}$ — translate vs. reflect across $x$-axis. The translation only aff

[5] #13 8.G.A.3 Pair $\{1,4\}$ — translate vs. dilate. Dilation from the origin scales the trans

[6] #13 8.G.A.3 Pair $\{2,3\}$ — rotate $90^{\circ}$ CCW vs. reflect across $x$-axis. Both touch

[7] #13 8.G.A.3 Pair $\{2,4\}$ — rotate $90^{\circ}$ CCW vs. dilate from origin. Both fix the or

[8] #13 6.EE.A.3 Pair $\{3,4\}$ — reflect across $x$-axis vs. dilate from origin. Reflection mult

[9] #2 4.OA.C.5 Count the commuting pairs from the list. From the six rows above, the ones marke

Review

Reasonableness: Spot-check each commuting pair on a concrete point. Take $(x,y) = (3, 5)$. Pair $\{1,3\}$: $T_3 \circ T_1$ gives $(5, -5)$ and $T_1 \circ T_3$ gives $(5, -5)$ — match. Pair $\{2,4\}$: $T_4 \circ T_2$ gives $(-10, 6)$ and $T_2 \circ T_4$ gives $(-10, 6)$ — match. Pair $\{3,4\}$: $T_4 \circ T_3$ gives $(6, -10)$ and $T_3 \circ T_4$ gives $(6, -10)$ — match. Spot-check a non-commuting pair, say $\{1,2\}$: $T_2 \circ T_1$ gives $(-5, 5)$ but $T_1 \circ T_2$ gives $(-3, 3)$ — clearly different. Numbers line up with the algebraic count of $3$.

Alternative: Tool #16 (Change Focus / Count the Complement): instead of asking which pairs commute, ask which involve the translation $T_1$ in a way that scales differently — translation is the lone non-origin-fixing move, so any pair $\{T_1, T_k\}$ with $k \in \{2, 4\}$ fails because rotation and dilation both move the translation vector; pair $\{T_1, T_3\}$ survives because reflection across the $x$-axis fixes the horizontal direction of the translation. That rules out $2$ of the $6$ pairs immediately. Among the three origin-fixing moves $\{T_2, T_3, T_4\}$, scaling ($T_4$) commutes with anything linear, while $T_2 \circ T_3$ vs. $T_3 \circ T_2$ gives two different reflections — so $\{2,3\}$ fails and $\{2,4\}, \{3,4\}$ succeed. Total commuting: $1 + 2 = 3$, matching (C).

CCSS standards used (min grade 8)

8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates (Translating each named transformation into a coordinate rule on $(x,y)$ — translation adds to $x$, rotation by $90^{\circ}$ CCW sends $(x,y) \to (-y,x)$, reflection across the $x$-axis flips $y$, dilation by factor $2$ scales both coordinates.)
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them) (Deciding whether the two composed coordinate expressions $(A(B(x,y)))$ and $(B(A(x,y)))$ agree for every $(x,y)$ — if the expressions are not identical, the pair does not commute.)
6.EE.A.3 Apply the properties of operations to generate equivalent expressions (Reordering scalar multiplications (e.g. $2 \cdot (-y) = -2y = -(2y)$) inside the coordinate composition to confirm equality for the commuting pairs.)
7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation (Enumerating the $\binom{4}{2} = 6$ unordered pairs of transformations in index order so that each pair is tested exactly once.)
4.OA.C.5 Generate a number or shape pattern that follows a given rule and identify apparent features (Tallying the entries marked "commute" in the systematic list to get the final count of $3$.)

⭐ When a problem asks how many out of a small collection satisfy a rule, list every case and decide each one — and once each geometric move is written as a coordinate formula, deciding "do they commute" is just checking whether two short expressions match.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 8)

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