Question

which type of transformation must have occurred in order to map triangle rst to its image, triangle rst?
a translation to the right
a translation to the left
a reflection over the y - axis
a clockwise rotation

Explanation:

Step1: Analyze translation

Translation moves a figure without rotation or reflection. Observing the coordinates of \(R(1, - 2)\) and \(R'(1,-2)\), \(S(3,3)\) and \(S'(3, - 3)\), \(T(3,1)\) and \(T'(3,-1)\). The \(x -\)coordinates remain the same, and the \(y -\)coordinates change sign. A translation to the right or left would change the \(x -\)coordinate. So it's not a translation to the right or left.

Step2: Analyze rotation

A clock - wise rotation would change both \(x\) and \(y\) coordinates in a non - symmetric (about \(x = 0\)) way. For a point \((x,y)\) rotated \(180^{\circ}\) clock - wise about the origin, the new point is \((-x,-y)\). But here \(x\) remains the same. So it's not a clock - wise rotation.

Step3: Analyze reflection

The rule for reflection over the \(x -\)axis is \((x,y)\to(x, - y)\). For reflection over the \(y -\)axis, the rule is \((x,y)\to(-x,y)\). Here, for each point \((x,y)\) in \(\triangle RST\) (e.g., \(R(1,-2)\), \(S(3,3)\), \(T(3,1)\)) and its image \((x,y')\) in \(\triangle R'S'T'\) (\(R'(1, - 2)\), \(S'(3,-3)\), \(T'(3,-1)\)), the transformation is \((x,y)\to(x, - y)\), which is a reflection over the \(x -\)axis.

Answer:

a reflection over the \(x -\)axis