Question

a robot travels from position a to b to c to d. what composition of rigid motions represents these moves?
assuming each movement is in a straight line, the composition ( t_{squaresquare} circ t_{squaresquare} circ t_{squaresquare} ) represents the moves.

Explanation:

Step1: Analyze A to B

First, determine the translation from A to B. Let's assume coordinates: A is at some point, B is a translation. Let's say A is \((x_1,y_1)\), B is \((x_2,y_2)\). From the grid, moving from A to B: let's check the change in x and y. Suppose A is \((-4,1)\), B is \((-2,3)\)? Wait, maybe better to look at the grid. Wait, the grid has x and y axes. Let's assume A is at \((-4,1)\), B is at \((-2,3)\)? No, maybe A to B: let's see the first translation. Wait, the problem is about rigid motions (translations, rotations, reflections). But since it's from A to B to C to D, and assuming translations (since it's a composition of \(T\) (translations)). Let's find the vectors for each move.

First, A to B: let's say A is at \((-4,1)\), B is at \((-2,3)\)? Wait, maybe the coordinates: Let's look at the grid. Let's assume the grid has x from -5 to 5, y from -2 to 6. A is at (-4,1), B is at (-2,3) (so translation vector (2,2)? Wait, no, maybe A to B: let's check the first translation. Wait, the problem is a composition of three translations? Wait, the composition is \(T_{(a,b)} \circ T_{(c,d)} \circ T_{(e,f)}\)? Wait, no, the problem says "the composition \(T_{\square\square} \circ T_{\square\square} \circ T_{\square\square}\) represents the moves". Wait, maybe the moves are A to B, B to C, C to D. So three translations: A→B, B→C, C→D.

Let's find the translation vectors:

1. A to B: Let's assume A is at (-4,1), B is at (-2,3). So the translation vector is (2,2) (change in x: -2 - (-4) = 2, change in y: 3 - 1 = 2). Wait, no, maybe A is at (-4,1), B is at (-2,3)? Wait, maybe the grid: Let's look at the points. A is at (-4,1), B is at (-2,3) (so translation (2,2)), B to C: C is at (2,3)? Wait, no, C is at (2,3)? Wait, D is at (3,5)? Wait, maybe I need to get the coordinates right.

Alternatively, let's look at the standard grid. Let's assume:

- A: (-4, 1)
- B: (-2, 3) (translation vector (2, 2))
- B to C: C is at (2, 3) (translation vector (4, 0))? No, wait, B to C: from (-2,3) to (2,3): translation vector (4,0)? No, that's horizontal. Then C to D: from (2,3) to (3,5): translation vector (1,2)? Wait, no, this is confusing. Wait, maybe the correct approach is to find the translation vectors for each segment:

1. A to B: Let's say the vector is (2, 2) (move 2 right, 2 up)
2. B to C: Let's say the vector is (4, 0) (move 4 right, 0 up)
3. C to D: Let's say the vector is (1, 2) (move 1 right, 2 up)

But wait, the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\)? No, composition of translations is done right to left (since \(T_{v} \circ T_{u}\) means first apply \(T_{u}\), then \(T_{v}\)). Wait, the order is important: \(T_{v} \circ T_{u}(P) = T_{v}(T_{u}(P)) = P + u + v\). So the composition of translations is the sum of the vectors.

But maybe the actual coordinates: Let's look at the grid again. Let's assume:

- A: (-4, 1)
- B: (-2, 3) (vector (2, 2))
- C: (2, 3) (vector (4, 0) from B: 2 - (-2) = 4, 3 - 3 = 0)
- D: (3, 5) (vector (1, 2) from C: 3 - 2 = 1, 5 - 3 = 2)

So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\)? Wait, no, the order is A→B (T1), B→C (T2), C→D (T3). So the composition is \(T3 \circ T2 \circ T1\), because to go from A to D, you do T1 (A→B), then T2 (B→C), then T3 (C→D). So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\)? Wait, no, the translation vectors:

T1 (A→B): vector (2, 2) (from (-4,1) to (-2,3): Δx=2, Δy=2)

T2 (B→C): vector (4, 0) (from (-2,3) to (2,3): Δx=4, Δy=0)

T3 (C→D): vector (1, 2) (from (2,3) to (3,5): Δx=1, Δy=2)

So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\). Wait, but maybe the coordinates are different. Alternatively, maybe the first translation (A→B) is (2,2), second (B→C) is (4,0), third (C→D) is (1,2). So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\). But maybe the actual vectors are:

Wait, let's check the grid again. Let's assume the points:

- A: (-4, 1)
- B: (-2, 3) (translation (2,2))
- C: (2, 3) (translation (4,0) from B: 2 - (-2) = 4, 3 - 3 = 0)
- D: (3, 5) (translation (1,2) from C: 3 - 2 = 1, 5 - 3 = 2)

So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\). But maybe the problem is simpler. Wait, maybe the first translation (A→B) is (2,2), second (B→C) is (4,0), third (C→D) is (1,2). So the answer would be the translation vectors for each \(T\).

Alternatively, maybe the coordinates are:

A: (-4, 1)

B: (-2, 3) → T(2,2)

B to C: (2, 3) → T(4,0) (from x=-2 to x=2: 4 units right, y same)

C to D: (3, 5) → T(1,2) (x from 2 to 3: 1 right, y from 3 to 5: 2 up)

So the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\).

Step2: Confirm the Translations

Each \(T_{(a,b)}\) represents a translation of \(a\) units in x and \(b\) units in y. So for A→B: (2,2), B→C: (4,0), C→D: (1,2). Thus, the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\).

Answer:

The composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\) (assuming the translation vectors are (2,2) for A→B, (4,0) for B→C, and (1,2) for C→D). So the boxes should be filled with (1,2), (4,0), (2,2) in order (since composition is rightmost first, then middle, then leftmost? Wait, no, composition order: \(T_v \circ T_u\) means apply \(T_u\) first, then \(T_v\). So to go from A to D, we do A→B (T1), B→C (T2), C→D (T3). So the composition is \(T3 \circ T2 \circ T1\), so the first \(T\) (leftmost) is T3, then T2, then T1? Wait, no, the notation \(T_{a,b} \circ T_{c,d}\) means first apply \(T_{c,d}\), then \(T_{a,b}\). So if we have three translations: T1 (A→B), T2 (B→C), T3 (C→D), then the composition is \(T3 \circ T2 \circ T1\), so the leftmost \(T\) is T3, middle is T2, rightmost is T1. So the vectors are T3: (1,2), T2: (4,0), T1: (2,2). Thus, the composition is \(T_{(1,2)} \circ T_{(4,0)} \circ T_{(2,2)}\).