Question

quadrilaterals mnpq and rstu are graphed on the coordinate plane. which transformation maps mnpq onto rstu? a 90° counterclockwise rotation about the origin a 180° counterclockwise rotation about the origin a reflection across the y - axis a translation 6 units to the left

Explanation:

Step1: Check translation

If we consider translation 6 units to the left. Let's take a point \(M(3,6)\). Translating 6 units left: \(x - 6\), \(y\) remains same. \(3-6=-3\), \(y = 6\). But in \(RSTU\) there is no point \((-3,6)\). So option D is wrong.

Step2: Check reflection across \(y -\)axis

The rule for reflection across \(y-\)axis is \((x,y)\to(-x,y)\). Take point \(M(3,6)\), after reflection \((-3,6)\). But in \(RSTU\) there is no point \((-3,6)\). So option C is wrong.

Step3: Check \(90^{\circ}\) counter - clockwise rotation

The rule for \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). Take point \(M(3,6)\), after rotation \((-6,3)\). But in \(RSTU\) there is no point \((-6,3)\). So option A is wrong.

Step4: Check \(180^{\circ}\) counter - clockwise rotation

The rule for \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\).
Take \(M(3,6)\to(-3, - 6)\) (not relevant). Take \(N(5,4)\to(-5,-4)\) (not relevant). Another way:
Let's use vector or shape orientation. A \(180^{\circ}\) rotation changes the orientation of the quadrilateral in a way that if we consider the relative positions of the vertices. For example, if we assume the center of rotation is the origin. The general property of \(180^{\circ}\) rotation: if we connect corresponding vertices of \(MNPQ\) and \(RSTU\) and find that the lines connecting them pass through the origin.

Answer:

B. a \(180^{\circ}\) counterclockwise rotation about the origin