Question

alicia analyzes the quadrilaterals shown on the coordinate plane. which equation could alicia use to show that these two shapes are congruent?
a. $(r_{(90^{circ},0)}circ t_{<-3,0>})(pqrs)=tuvw$
b. $(r_{(90^{circ},0)}circ t_{<0,-3>})(pqrs)=tuvw$
c. $(r_{(180^{circ},0)}circ t_{<-3,0>})(pqrs)=tuvw$
d. $(r_{(180^{circ},0)}circ t_{<0,-3>})(pqrs)=tuvw$

Explanation:

Step1: Analyze rotation

A \(180^{\circ}\) rotation about the origin \((r_{(180^{\circ},O)})\) changes the sign of both \(x\) and \(y\) coordinates of a point \((x,y)\) to \((-x,-y)\). A \(90^{\circ}\) rotation does not map the quadrilateral \(PQRS\) to \(TUVW\) in the correct orientation.

Step2: Analyze translation

After a \(180^{\circ}\) rotation, we need to check the translation. If we consider the movement of a point (say \(P( - 1,3)\) in \(PQRS\)). After \(180^{\circ}\) rotation about the origin, it becomes \((1, - 3)\). To get to the corresponding point \(W(-1,0)\) in \(TUVW\), we need a translation \(T_{\langle - 3,0
angle}\) (because \(1-3=-2\) ( \(x\) - coordinate) and \(-3 + 0=-3\) ( \(y\) - coordinate is not relevant here as we are just checking the composition).

Answer:

C. \((r_{(180^{\circ},O)}\circ T_{\langle - 3,0
angle})(PQRS)=TUVW\)