Question

the coordinates of the vertices of △ rst are r(-3, -1), s(-1, -1), and t(-4, -5). the coordinates of the vertices of △ rst are r(1, -2), s(1, 0), and t(5, -3). what is the sequence of transformations that maps △ rst to △ rst? drag and drop the answers into the boxes to correctly complete the statement. a sequence of transformations that maps △ rst to △ rst is a first transformation followed by a second transformation. options: translation 1 unit up, rotation of 180° about the origin, rotation of 90° counterclockwise about the origin, reflection across the y-axis.

Explanation:

Step1: Analyze rotation of 180° about origin

A 180° rotation about the origin transforms a point \((x,y)\) to \((-x,-y)\). Let's test this on point \(R(-3,-1)\): applying 180° rotation, we get \((3,1)\). Now let's check the next transformation.

Step2: Analyze translation 1 unit down? Wait, no, let's check the given options. Wait, maybe first rotation 180° then translation? Wait, no, let's check the coordinates. Wait, let's take point \(R(-3,-1)\), after rotation 180°: \((3,1)\). Then if we translate 1 unit down? No, the final \(R'\) is \((1,-2)\). Wait, maybe another approach. Wait, let's check reflection across y - axis: reflection across y - axis transforms \((x,y)\) to \((-x,y)\). For \(R(-3,-1)\), reflection across y - axis gives \((3,-1)\). Then translation? Wait, \(R'\) is \((1,-2)\). Wait, maybe rotation 180° first. Wait, \(R(-3,-1)\) rotated 180°: \((3,1)\). Then translation: from \((3,1)\) to \((1,-2)\): the change in x is \(1 - 3=-2\), change in y is \(-2 - 1=-3\). No, that's not matching. Wait, maybe rotation 90° counterclockwise? Rotation 90° counterclockwise about origin: \((x,y)\to(-y,x)\). For \(R(-3,-1)\): \((1,-3)\). Not matching \(R'(1,-2)\). Wait, let's check the y - coordinate of \(R\) and \(R'\): \(R\) has \(y=-1\), \(R'\) has \(y = - 2\), difference of - 1 (1 unit down? But option is 1 unit up. Wait, maybe I made a mistake. Wait, let's check the vectors. Let's look at \(S(-1,-1)\) and \(S'(1,0)\). Let's try rotation 180° on \(S(-1,-1)\): \((1,1)\). Then translation 1 unit down: \((1,0)\), which matches \(S'\). For \(T(-4,-5)\) rotated 180°: \((4,5)\). Then translation 1 unit down: \((4,4)\)? No, \(T'\) is \((5,-3)\). Wait, no, maybe first translation? Wait, the options are rotation 180°, rotation 90° counterclockwise, reflection across y - axis, translation 1 unit up. Let's try reflection across y - axis on \(R(-3,-1)\): \((3,-1)\). Then translation 1 unit down? No, option is 1 unit up. Wait, \((3,-1)\) translated 1 unit up: \((3,0)\), not \(R'(1,-2)\). Wait, let's try rotation 180° on \(R(-3,-1)\): \((3,1)\). Then translation 1 unit down: \((3,0)\), no. Wait, maybe the first transformation is rotation 180° about the origin, then translation? Wait, no, let's check the correct sequence. Let's take \(R(-3,-1)\), \(S(-1,-1)\), \(T(-4,-5)\). Let's apply rotation 180° about origin: \(R_1=(3,1)\), \(S_1=(1,1)\), \(T_1=(4,5)\). Then apply translation: from \(R_1=(3,1)\) to \(R'=(1,-2)\): the translation vector is \((1 - 3,-2 - 1)=(-2,-3)\). But that's not one of the options. Wait, maybe the first transformation is reflection across y - axis? \(R(-3,-1)\) reflected across y - axis: \((3,-1)\). \(S(-1,-1)\) reflected across y - axis: \((1,-1)\). \(T(-4,-5)\) reflected across y - axis: \((4,-5)\). Then translation 1 unit up: \(R_2=(3,0)\), \(S_2=(1,0)\), \(T_2=(4,-4)\). No, \(T'\) is \((5,-3)\). Wait, maybe rotation 90° counterclockwise? \(R(-3,-1)\) rotated 90° counterclockwise: \((1,-3)\). \(S(-1,-1)\) rotated 90° counterclockwise: \((1,-1)\). \(T(-4,-5)\) rotated 90° counterclockwise: \((5,-4)\). Then translation 1 unit up: \(R_2=(1,-2)\), \(S_2=(1,0)\), \(T_2=(5,-3)\). Yes! That matches. So first, rotation of 90° counterclockwise about the origin, then translation 1 unit up. Wait, but the options are: first box and then second box. Wait, let's check again. Wait, rotation 90° counterclockwise about origin: \((x,y)\to(-y,x)\). For \(R(-3,-1)\): \(-y = 1\), \(x=-3\)? Wait, no, rotation 90° counterclockwise about origin is \((x,y)\to(-y,x)\). So \(R(-3,-1)\): \(x=-3\), \(y = - 1\), so \(-y = 1\), \(x=-3\)? No, that's wrong. Wait, rotation 90° counterclockwise: \((x,y)\) becomes \((-y,x)\). So \(R(-3,-1)\): \(-y = 1\), \(x=-3\)? No, \((-3,-1)\) rotated 90° counterclockwise: \((1,-3)\). Let's calculate: the formula for 90° counterclockwise rotation about origin is \((x,y)\mapsto(-y,x)\). So \(x=-3\), \(y=-1\), so \(-y = 1\), \(x=-3\)? No, that's \((1,-3)\). Yes, because \(-y=1\), \(x = - 3\)? Wait, no, the formula is \((x,y)\to(-y,x)\). So \(x=-3\), \(y=-1\), so \(-y = 1\), \(x=-3\)? No, that's \((1,-3)\). Then translation 1 unit up: \((1,-3 + 1)=(1,-2)\), which is \(R'\). For \(S(-1,-1)\): rotated 90° counterclockwise: \((1,-1)\), then translation 1 unit up: \((1,0)\), which is \(S'\). For \(T(-4,-5)\): rotated 90° counterclockwise: \((5,-4)\), then translation 1 unit up: \((5,-3)\), which is \(T'\). So the sequence is rotation of 90° counterclockwise about the origin followed by translation 1 unit up. Wait, but the options are: first box: rotation of 180° about the origin, rotation of 90° counterclockwise about the origin, reflection across the y - axis; second box: translation 1 unit up, etc. So first transformation: rotation of 90° counterclockwise about the origin, second transformation: translation 1 unit up. Wait, but let's check another way. Wait, maybe I made a mistake in rotation. Let's check reflection across y - axis: \(R(-3,-1)\) becomes \((3,-1)\), then translation: \((3,-1)\) to \((1,-2)\): change in x: \(1 - 3=-2\), change in y: \(-2-(-1)=-1\) (1 unit down), but option is 1 unit up. No. Rotation 180°: \(R(-3,-1)\) becomes \((3,1)\), then translation: \((3,1)\) to \((1,-2)\): change in x: - 2, change in y: - 3. Not matching. So the correct sequence is rotation of 90° counterclockwise about the origin followed by translation 1 unit up.

Answer:

A sequence of transformations that maps \(\triangle RST\) to \(\triangle R'S'T'\) is a \(\boldsymbol{\text{rotation of } 90^\circ \text{ counterclockwise about the origin}}\) followed by a \(\boldsymbol{\text{translation 1 unit up}}\)