Question

triangle uvw is the image of triangle uvw under a rotation 180° about the origin followed by a translation.
write the rules for the rotation and translation.
rotation: (x, y) ↦ (\boxed{ }, \boxed{ })
translation: (x, y) ↦ (\boxed{ }, \boxed{ })

Explanation:

Step1: Recall 180° rotation rule

A 180° rotation about the origin transforms a point \((x, y)\) to \((-x, -y)\). So the rotation rule is \((x, y) \mapsto (-x, -y)\).

Step2: Find translation using a point

Take point \(U(2, 4)\). After 180° rotation, it becomes \((-2, -4)\). The final point \(U'\) is \((8, -2)\). Let the translation vector be \((h, k)\), so \(-2 + h = 8\) and \(-4 + k = -2\). Solving for \(h\): \(h = 8 + 2 = 10\)? Wait, no, wait. Wait, maybe take point \(W(4, 2)\). After rotation: \((-4, -2)\). \(W'\) is \((6, 1)\). So \(-4 + h = 6\) ⇒ \(h = 10\)? No, wait, maybe I made a mistake. Wait, let's take point \(V(8, 6)\). After 180° rotation: \((-8, -6)\). \(V'\) is \((2, -3)\). So \(-8 + h = 2\) ⇒ \(h = 10\), and \(-6 + k = -3\) ⇒ \(k = 3\). Wait, but let's check \(U\): \(U(2, 4)\) rotated 180° is \((-2, -4)\). Then translate by \((10, 3)\): \(-2 + 10 = 8\), \(-4 + 3 = -1\)? No, \(U'\) is \((8, -2)\). Wait, maybe I messed up the rotation. Wait, no, the rotation is 180° about origin, then translation. Let's take \(W(4, 2)\). Rotated 180°: \((-4, -2)\). Then \(W'\) is \((6, 1)\). So the translation is \((6 - (-4), 1 - (-2)) = (10, 3)\)? Wait, 6 - (-4) = 10, 1 - (-2) = 3. Then \(U(2, 4)\) rotated: \((-2, -4)\). Translated by (10, 3): \(-2 + 10 = 8\), \(-4 + 3 = -1\). But \(U'\) is (8, -2). Hmm, maybe I took the wrong point. Wait, \(U\) is (2, 4)? Wait, looking at the graph: \(U\) is at (2, 4)? Wait, the grid: x=2, y=4? Yes. \(U'\) is at (8, -2). So after rotation, let's say \(U\) becomes \(U_{rot}\), then \(U_{rot} + (h, k) = U'\). Let \(U_{rot} = (-2, -4)\) (180° rotation). Then \(-2 + h = 8\) ⇒ \(h = 10\), \(-4 + k = -2\) ⇒ \(k = 2\). Ah, there we go. So \(k = 2\). Let's check \(W\): \(W(4, 2)\) rotated: \((-4, -2)\). Then \(-4 + 10 = 6\), \(-2 + 2 = 0\)? No, \(W'\) is (6, 1). Wait, no, \(W'\) is (6, 1). So \(-4 + h = 6\) ⇒ \(h = 10\), \(-2 + k = 1\) ⇒ \(k = 3\). But \(U\): \(-2 + 10 = 8\), \(-4 + 3 = -1\), but \(U'\) is (8, -2). So maybe my initial point for \(U\) is wrong. Wait, looking at the graph: \(U\) is at (2, 4)? Wait, the x-axis: from -10 to 10, y-axis from -10 to 10. \(U\) is at (2, 4)? Yes. \(U'\) is at (8, -2). So the rotation is 180°: \((x,y)→(-x,-y)\). Then translation: let's find the vector from rotated point to final point. For \(U\): rotated \(U\) is \((-2, -4)\), final \(U'\) is \((8, -2)\). So translation vector is \((8 - (-2), -2 - (-4)) = (10, 2)\). Let's check \(W\): \(W(4, 2)\) rotated is \((-4, -2)\). Translated by (10, 2): \(-4 + 10 = 6\), \(-2 + 2 = 0\)? No, \(W'\) is (6, 1). Wait, \(W'\) is (6, 1). So \(-4 + 10 = 6\), \(-2 + 3 = 1\). Ah, so \(k = 3\) for \(W\), \(k = 2\) for \(U\). Wait, maybe I misread the coordinates. Let's re - check the graph:

- \(U\): (2, 4)
- \(U'\): (8, -2)
- \(V\): (8, 6)
- \(V'\): (2, -3)
- \(W\): (4, 2)
- \(W'\): (6, 1)

Now, apply 180° rotation to \(U(2,4)\): \((-2, -4)\). Then to get to \(U'(8, -2)\), the translation is \(8 - (-2)=10\) (x - direction) and \(-2 - (-4)=2\) (y - direction). Wait, but for \(V(8,6)\) rotated: \((-8, -6)\). To get to \(V'(2, -3)\): \(2 - (-8)=10\) (x - direction), \(-3 - (-6)=3\) (y - direction). For \(W(4,2)\) rotated: \((-4, -2)\). To get to \(W'(6,1)\): \(6 - (-4)=10\) (x - direction), \(1 - (-2)=3\) (y - direction). Wait, there's a discrepancy in y - direction for \(U\). Wait, maybe I misread \(U\)'s coordinates. Wait, looking at the graph, \(U\) is at (2, 4)? Wait, no, maybe \(U\) is at (2, 4)? Wait, the \(U'\) is at (8, -2). Let's recalculate \(U\)'s rotation and translation. If rotation is \((x,y)→(-x,-y)\), then \(U(2,4)\) becomes \((-2, -4)\). Then translation: \(8 - (-2)=10\), \(-2 - (-4)=2\). But for \(W(4,2)\), rotation gives \((-4, -2)\), translation gives \(6 - (-4)=10\), \(1 - (-2)=3\). For \(V(8,6)\), rotation gives \((-8, -6)\), translation gives \(2 - (-8)=10\), \(-3 - (-6)=3\). So maybe \(U\)'s y - coordinate is misread. Wait, maybe \(U\) is at (2, 4), and \(U'\) is at (8, -2). Then the y - translation for \(U\) is \(-2 - (-4)=2\), but for \(W\) and \(V\) it's 3. That can't be. Wait, maybe the rotation is not 180° about origin? No, the problem says 180° about origin. Wait, maybe I made a mistake in the rotation rule. Wait, 180° rotation about origin: \((x,y)→(-x,-y)\) is correct. Let's check \(V(8,6)\): rotated is \((-8, -6)\). \(V'\) is (2, -3). So \(-8 + 10 = 2\), \(-6 + 3 = -3\). Correct. \(W(4,2)\): rotated \((-4, -2)\), \(-4 + 10 = 6\), \(-2 + 3 = 1\). Correct. \(U(2,4)\): rotated \((-2, -4)\), \(-2 + 10 = 8\), \(-4 + 3 = -1\). But \(U'\) is (8, -2). So maybe \(U\) is at (2, 3)? No, the graph shows \(U\) at (2, 4). Wait, maybe the problem has a typo, but according to the majority (V and W), the translation in y is 3, and x is 10. So maybe I misread \(U'\)'s y - coordinate. Let's check the graph again: \(U'\) is at (8, -2)? Wait, no, looking at the graph, \(U'\) is at (8, -2)? Wait, the blue triangle: \(U'\) is at (8, -2), \(V'\) at (2, -3), \(W'\) at (6, 1). So for \(U(2,4)\), rotation: \((-2, -4)\), translation: (10, 3) gives \(-2 + 10 = 8\), \(-4 + 3 = -1\), but \(U'\) is (8, -2). There's a mistake here. Wait, maybe the rotation is 180° about a different point? No, the problem says about origin. Wait, maybe I swapped x and y. No, 180° rotation is \((-x, -y)\). Wait, let's take \(U(2, 4)\), \(U'(8, -2)\). The vector from \(U\) to \(U'\) is (6, -6). A 180° rotation followed by translation. Let the rotation be \(R\) and translation be \(T\), so \(T(R(U)) = U'\). So \(R(U)=U' - T\). If \(R(U)=(-2, -4)\), then \(U' - T = (-2, -4)\) ⇒ \(T = U' - (-2, -4)=(8 + 2, -2 + 4)=(10, 2)\). But for \(V(8,6)\), \(R(V)=(-8, -6)\), \(T = V' - (-8, -6)=(2 + 8, -3 + 6)=(10, 3)\). This is a problem. Wait, maybe the initial triangle's coordinates are different. Wait, looking at the graph, \(W\) is at (4, 2), \(W'\) at (6, 1). So \(R(W)=(-4, -2)\), \(T = W' - (-4, -2)=(6 + 4, 1 + 2)=(10, 3)\). \(V(8,6)\): \(R(V)=(-8, -6)\), \(T=(2 + 8, -3 + 6)=(10, 3)\). \(U\): let's re - check \(U\)'s coordinates. Maybe \(U\) is at (2, 3)? No, the blue dot for \(U\) is at (2, 4). Wait, maybe the \(U'\) is at (8, -1). Then \(-4 + 3 = -1\), which matches. Maybe a typo in the graph. So assuming the translation is (10, 3) for \(V\) and \(W\), so the rotation rule is \((x,y)→(-x, -y)\), and translation is \((x + 10, y + 3)\)? Wait, no, translation is \((h, k)\), so \((x,y)→(x + h, y + k)\). So after rotation \((-x, -y)\), then translation: \((-x + h, -y + k)\). So for \(W(4,2)\): \(-4 + h = 6\) ⇒ \(h = 10\), \(-2 + k = 1\) ⇒ \(k = 3\). So the translation rule is \((x, y)→(x + 10, y + 3)\)? Wait, no, the translation is applied after rotation, so the rotation is \((x,y)→(-x, -y)\), then translation \((x,y)→(x + 10, y + 3)\)? Wait, no, the translation is a vector \((h, k)\), so the combined transformation is \((x,y)→(-x + h, -y + k)\). So for \(W\): \(-4 + h = 6\) ⇒ \(h = 10\), \(-2 + k = 1\) ⇒ \(k = 3\). So the translation rule is \((x, y)→(x + 10, y + 3)\)? Wait, no, the translation is \((h, k)\), so when we translate a point \((a, b)\), we get \((a + h, b + k)\). So after rotation, the point is \((-x, -y)\), then we translate it by \((h, k)\) to get \((-x + h, -y + k)\). So for \(W\): \(-4 + h = 6\) ⇒ \(h = 10\), \(-2 + k = 1\) ⇒ \(k = 3\). So the translation rule is \((x, y)→(x + 10, y + 3)\)? Wait, no, the translation is a function that takes the rotated point and adds (h, k). So the translation rule is \((x, y)→(x + 10, y + 3)\)? Wait, let's confirm with \(V\): \(V(8,6)\) rotated is \((-8, -6)\), then translated by (10, 3): \(-8 + 10 = 2\), \(-6 + 3 = -3\), which matches \(V'(2, -3)\). \(W(4,2)\) rotated is \((-4, -2)\), translated by (10, 3): \(-4 + 10 = 6\), \(-2 + 3 = 1\), which matches \(W'(6, 1)\). \(U(2,4)\) rotated is \((-2, -4)\), translated by (10, 3): \(-2 + 10 = 8\), \(-4 + 3 = -1\). But \(U'\) is at (8, -2). So maybe \(U\) is at (2, 3). Then rotated: \((-2, -3)\), translated by (10, 3): \(-2 + 10 = 8\), \(-3 + 3 = 0\). No, still not. Wait, maybe the rotation is 180° about the origin, then the translation is (10, 3) for \(V\) and \(W\), and maybe \(U\)'s coordinate is misread. So we'll go with the majority (V and W) for the translation.

Answer:

Rotation: \((x, y) \mapsto (-x, -y)\)
Translation: \((x, y) \mapsto (x + 10, y + 3)\)