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 box followed by a second box. the options are: 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 (90° counterclockwise)

A 90° counterclockwise rotation about the origin transforms a point \((x,y)\) to \((-y,x)\). Let's test on \(R(-3,-1)\): \((-(-1), -3)=(1, -3)\)? No, wait, maybe first check rotation. Wait, let's check the other transformation. Wait, maybe first do rotation 90° counterclockwise. Let's take \(R(-3,-1)\): 90° counterclockwise rotation: \((x,y)\to(-y,x)\), so \((-(-1), -3)=(1, -3)\). Then \(S(-1,-1)\): \((-(-1), -1)=(1, -1)\). \(T(-4,-5)\): \((-(-5), -4)=(5, -4)\). Now, compare with \(R'(1,-2)\), \(S'(1,0)\), \(T'(5,-3)\). The difference in y - coordinates: \(R\) after rotation: \(y=-3\), \(R'\) has \(y=-2\): so translation 1 unit up (\(y\to y + 1\)). Let's check: After rotation 90° counterclockwise, \(R(1,-3)\) + translation 1 unit up: \((1, -3 + 1)=(1, -2)\) (matches \(R'\)). \(S(1,-1)\) + translation 1 unit up: \((1, -1 + 1)=(1, 0)\) (matches \(S'\)). \(T(5,-4)\) + translation 1 unit up: \((5, -4 + 1)=(5, -3)\) (matches \(T'\)). So first rotation 90° counterclockwise about the origin, then translation 1 unit up.

Step2: Confirm the sequence

First, apply rotation of 90° counterclockwise about the origin to \(\triangle RST\), then apply 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 \(\boldsymbol{\text{translation 1 unit up}}\)