Question

which sequence of transformations maps rst onto rst? a reflection across the x - axis followed by a rotation 90° clockwise around the origin a rotation 180° around the origin followed by a translation left 3 units and up 4 units a translation right 3 units and up 10 units followed by a reflection across the y - axis

Explanation:

Step1: Analyze reflection across x - axis

A reflection across the x - axis changes the sign of the y - coordinate of each point. Let's assume a general point \((x,y)\) becomes \((x, - y)\).

Step2: Analyze 90 - degree clockwise rotation around origin

A 90 - degree clockwise rotation around the origin transforms a point \((x,y)\) to \((y,-x)\).
Let's take a point from \(\triangle RST\), say \(R(-3,-3)\). After reflection across the x - axis, \(R\) becomes \((-3,3)\). After a 90 - degree clockwise rotation around the origin, \((-3,3)\) becomes \((3,3)\) which is not the correct location of \(R'\) in \(\triangle R'S'T'\).

Step3: Analyze 180 - degree rotation around origin

A 180 - degree rotation around the origin changes a point \((x,y)\) to \((-x,-y)\). For \(R(-3,-3)\), it becomes \((3,3)\).

Step4: Analyze translation left 3 units and up 4 units

A translation left 3 units and up 4 units on a point \((x,y)\) gives \((x - 3,y + 4)\). For the point \((3,3)\) after 180 - degree rotation, \((3-3,3 + 4)=(0,7)\) which is not the location of \(R'\).

Step5: Analyze translation right 3 units and up 10 units

For a point \((x,y)\) of \(\triangle RST\), say \(R(-3,-3)\), after translation right 3 units and up 10 units, it becomes \((-3+3,-3 + 10)=(0,7)\).

Step6: Analyze reflection across y - axis

A reflection across the y - axis changes a point \((x,y)\) to \((-x,y)\). For the point \((0,7)\) after translation, it remains \((0,7)\) (since \(x = 0\)).
Let's check with another point \(S(-6,-5)\). After translation right 3 units and up 10 units, \(S\) becomes \((-6 + 3,-5+10)=(-3,5)\). After reflection across the y - axis, it becomes \((3,5)\) which is the location of \(S'\).

Answer:

a translation right 3 units and up 10 units followed by a reflection across the y - axis