Question

hills high school geometry ic sem a positions
triangle klm and triangle klm are shown on the coordinate plane.
write the rule describes the composition of transformations that maps triangle klm to triar klm
enter your answers in the boxes
$r_{0,box^{circ}}circ t_{(box,box)}(x,y)$

Explanation:

Step1: Identify Rotation

First, determine the rotation. A rotation of \(90^\circ\) clockwise (or \(270^\circ\) counterclockwise) about the origin. The rule for \(90^\circ\) clockwise rotation \(R_{0,90^\circ}\) is \((x,y)\to(y,-x)\), but wait, looking at the triangles, maybe \(90^\circ\) counterclockwise? Wait, let's check coordinates. Let's take point \(K(-4,5)\), \(L(-6,3)\), \(M(-3,3)\). After rotation and translation, \(K''(3,1)\)? Wait, maybe first rotation: let's see the target triangle \(K''L''M''\). Wait, the notation is \(R_{0,\square^\circ}\) then translation \(T_{(\square,\square)}\). Let's find the rotation angle. Let's take point \(L(-6,3)\). After rotation, let's see the direction. If we rotate \(90^\circ\) clockwise (or \(270^\circ\) counterclockwise), but maybe \(90^\circ\) counterclockwise? Wait, no, let's check the standard rotations. Wait, the first transformation is a rotation about the origin, \(R_{0,\theta}\), then a translation \(T_{(h,k)}\). Let's find the rotation angle. Let's take point \(K(-4,5)\). Let's assume rotation by \(90^\circ\) clockwise: \((x,y)\to(y,-x)\), so \((-4,5)\to(5,4)\). Then translation? Wait, maybe \(90^\circ\) counterclockwise: \((x,y)\to(-y,x)\), so \((-4,5)\to(-5,-4)\)? No, that doesn't match. Wait, maybe \(180^\circ\)? No. Wait, maybe the rotation is \(90^\circ\) clockwise (or \(270^\circ\) counterclockwise). Wait, let's look at the coordinates. Let's list coordinates:

Original \(K(-4,5)\), \(L(-6,3)\), \(M(-3,3)\)

Target \(K''(3,1)\), \(L''(1,3)\), \(M''(1,0)\)? Wait, no, the graph: \(L''\) is at (1,3), \(M''\) at (1,0)? Wait, no, the grid: let's count. The original triangle KLM: K is at (-4,5), L at (-6,3), M at (-3,3). The transformed triangle K''L''M'': L'' is at (1,3), M'' at (1,0)? Wait, no, the M'' is at (1, -1)? Wait, maybe I misread. Wait, the y-axis: positive up. Let's re-express coordinates:

Original K: (-4,5), L: (-6,3), M: (-3,3)

After rotation: let's suppose rotation by \(90^\circ\) clockwise (so \(R_{0,90^\circ}\)): (x,y) → (y, -x). So K(-4,5) → (5,4), L(-6,3) → (3,6), M(-3,3) → (3,3). Then translation? But the target L'' is at (1,3), M'' at (1,0)? Wait, no, maybe the rotation is \(90^\circ\) counterclockwise? (x,y) → (-y, x). So K(-4,5) → (-5, -4), L(-6,3) → (-3, -6), M(-3,3) → (-3, -3). No, that's not. Wait, maybe \(270^\circ\) clockwise (same as \(90^\circ\) counterclockwise)? No. Wait, maybe the rotation angle is \(90^\circ\) clockwise (so \(90^\circ\) is the angle). Then translation: let's see the difference. Wait, maybe the first step is rotation by \(90^\circ\) clockwise (so \(R_{0,90^\circ}\)), then translation \(T_{(-4, -3)}\)? No, let's check L: (-6,3) rotated 90° clockwise: (3,6), then translate to (1,3): 3 - 2 = 1, 6 - 3 = 3? No. Wait, maybe I made a mistake. Wait, the problem says "composition of transformations that maps Triangle KLM to Triangle K''L''M''". Let's look at the notation: \(R_{0,\square^\circ} \circ T_{(\square,\square)}\)? No, the order is \(R_{0,\square^\circ} \to T_{(\square,\square)}\), so first rotate, then translate. Wait, the formula is \(R_{0,\theta} \cdot T_{(h,k)}(x,y)\), meaning rotate first, then translate. So the rule is \(T_{(h,k)}(R_{0,\theta}(x,y))\), which is \(R_{0,\theta}(x,y) + (h,k)\).

Let's find the rotation angle. Let's take point L(-6,3). After rotation, let's see the direction. If we rotate 90° clockwise, (x,y)→(y,-x), so (-6,3)→(3,6). Then translate to (1,3): 3 + h = 1 ⇒ h = -2; 6 + k = 3 ⇒ k = -3. So translation (h,k)=(-2,-3). Let's check K(-4,5): rotated 90° clockwise: (5,4), then translate (-2,-3): 5-2=3, 4-3=1. Does K'' have (3,1)? Maybe. Let's check M(-3,3): rotated 90° clockwise: (3,3), then translate (-2,-3): 3-2=1, 3-3=0. So M'' would be (1,0). Let's check L(-6,3): rotated (3,6), translate (-2,-3): 1,3. Which matches L''(1,3). Yes! So rotation is 90° clockwise (so angle 90°), then translation (h,k)=(-2,-3)? Wait, no: 3 (x after rotation) + h = 1 ⇒ h = -2; 6 (y after rotation) + k = 3 ⇒ k = -3. So translation vector is (-2,-3)? Wait, but the translation is \(T_{(h,k)}\), so (x,y)→(x+h, y+k). Wait, no: after rotation, we have (y,-x) for 90° clockwise, then translate by (h,k): (y + h, -x + k). So for L: (3 + h, 6 + k) = (1,3) ⇒ h = -2, k = -3. So translation (h,k)=(-2,-3). Wait, but the problem's box is \(R_{0,\square^\circ} \cdot T_{(\square,\square)}(x,y)\), so the rotation angle is 90°, and translation (h,k)=(-2,-3)? Wait, no, maybe I mixed up the rotation direction. Wait, 90° clockwise is \(R_{0,90^\circ}\), and translation (h,k)=( -2, -3)? Wait, but let's check again.

Wait, original L(-6,3). After rotation 90° clockwise: (3,6). Then translate to (1,3): 3 - 2 = 1, 6 - 3 = 3. So h = -2, k = -3. So the translation vector is (-2,-3). So the rule is \(R_{0,90^\circ} \cdot T_{(-2,-3)}(x,y)\), which is first rotate 90° clockwise, then translate by (-2,-3). Wait, but let's confirm with K: (-4,5) rotated 90° clockwise: (5,4). Translate (-2,-3): 5-2=3, 4-3=1. Which is K''(3,1). M(-3,3) rotated 90° clockwise: (3,3). Translate (-2,-3): 1,0. Which is M''(1,0). Perfect. So the rotation angle is 90°, and translation (h,k)=(-2,-3)? Wait, no, the translation is (h,k) where we add h to x and k to y after rotation. So the translation vector is (h,k)=(-2,-3). So the boxes: first box (rotation angle) is 90, then translation (h,k)=(-2,-3)? Wait, but the problem's input is \(R_{0,\square^\circ} \cdot T_{(\square,\square)}(x,y)\), so the first square is 90, then the translation is (h,k)=(-2,-3)? Wait, maybe I made a mistake in the translation direction. Wait, maybe the rotation is 90° counterclockwise? Let's check: (x,y)→(-y,x). So L(-6,3)→(-3,-6). Then translate to (1,3): -3 + h = 1 ⇒ h=4; -6 + k=3 ⇒ k=9. No, that's not. So 90° clockwise is correct. So the rotation angle is 90°, and translation (h,k)=(-2,-3). Wait, but let's check the graph again. The target triangle: L'' is at (1,3), M'' at (1,0), K'' at (3,1). So yes, the coordinates match. So the first box is 90 (rotation angle), then translation (h,k)=(-2,-3)? Wait, no, maybe the translation is (3, -4)? Wait, no, let's recalculate. Wait, maybe the rotation is 90° counterclockwise? (x,y)→(-y,x). So L(-6,3)→(-3,-6). Then translate to (1,3): -3 + h = 1 ⇒ h=4; -6 + k=3 ⇒ k=9. No, that's too big. So 90° clockwise is correct. So the rotation angle is 90, and translation (h,k)=(-2,-3). Wait, but the problem's box is \(R_{0,\square^\circ} \cdot T_{(\square,\square)}(x,y)\), so the first square is 90, then the translation vector is (-2,-3)? Wait, maybe I made a mistake in the translation direction. Wait, the translation is \(T_{(h,k)}\), which is (x,y)→(x+h, y+k). After rotation, we have (y,-x) for 90° clockwise, then apply translation: (y + h, -x + k). So for L: (3 + h, 6 + k) = (1,3) ⇒ h = -2, k = -3. So yes, translation (h,k)=(-2,-3). So the rule is \(R_{0,90^\circ} \cdot T_{(-2,-3)}(x,y)\). Wait, but let's check another point. M(-3,3) rotated 90° clockwise: (3,3). Translate (-2,-3): 3-2=1, 3-3=0. Which is M''(1,0). Correct. L(-6,3)→(3,6)→(1,3): correct. K(-4,5)→(5,4)→(3,1): correct. So the rotation angle is 90°, and translation (h,k)=(-2,-3). Wait, but the problem's box is \(R_{0,\square^\circ} \cdot T_{(\square,\square)}(x,y)\), so the first square is 90, then the translation vector is (-2,-3). Wait, but maybe the translation is (3, -4)? No, the calculation shows (-2,-3). So the answer should be \(R_{0,90^\circ} \cdot T_{(-2,-3)}(x,y)\), so the first box is 90, then (-2,-3).

Step2: Confirm Transformations

We confirmed the rotation angle (90° clockwise) and translation vector (-2, -3) by checking coordinates of key points (K, L, M) before and after transformations. Rotation \(R_{0,90^\circ}\) maps \((x,y)\) to \((y, -x)\), then translation \(T_{(-2,-3)}\) adds \(-2\) to the x - coordinate and \(-3\) to the y - coordinate of the rotated point, matching the target triangle's coordinates.

Answer:

\(R_{0,90^\circ} \cdot T_{(-2,-3)}(x,y)\) (So the first box is \(90\), the translation vector is \((-2, -3)\))