Question

determining if figures are congruent
are δdef and δrpq congruent?
○ yes. δdef can be mapped to δrpq by a reflection across the x - axis followed by a translation 2 units left.
○ yes. δdef can be mapped to δrpq by a 180° rotation about the origin followed by a translation 2 units down.
○ no. δdef cannot be mapped to δrpq by a rigid transformation or combination of rigid transformations.

Explanation:

Step1: Analyze Reflection Across x - axis

First, consider reflecting \(\triangle DEF\) across the \(x\) - axis. The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). For point \(D(1,3)\) (assuming the coordinates of \(D\) are \((1,3)\), \(F(1,0)\), \(E(5,0)\)), after reflection across the \(x\) - axis, \(D\) becomes \((1,-3)\), \(F\) remains \((1,0)\), and \(E\) remains \((5,0)\). Then, translating 2 units left: the rule for translating a point \((x,y)\) \(h\) units left is \((x,y)\to(x - h,y)\). So, \((1,-3)\to(1 - 2,-3)=(-1,-3)\), \((1,0)\to(1 - 2,0)=(-1,0)\), \((5,0)\to(5 - 2,0)=(3,0)\). But the coordinates of \(\triangle RPQ\) are \(R(-1,-5)\), \(Q(-1,-2)\), \(P(-5,-2)\) (from the graph). This doesn't match, so let's check the first option's logic again. Wait, maybe my coordinate assumption is wrong. Let's re - identify the coordinates. From the graph: \(D\) is at \((1,3)\), \(F\) at \((1,0)\), \(E\) at \((5,0)\). \(\triangle RPQ\): \(R(-1,-5)\), \(Q(-1,-2)\), \(P(-5,-2)\).

Step2: Analyze 180 - degree Rotation

The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). For \(D(1,3)\), after \(180^{\circ}\) rotation: \((-1,-3)\). For \(F(1,0)\): \((-1,0)\). For \(E(5,0)\): \((-5,0)\). Then translating 2 units down: \((x,y)\to(x,y - 2)\). So, \((-1,-3)\to(-1,-5)\) (matches \(R\)), \((-1,0)\to(-1,-2)\) (matches \(Q\)), \((-5,0)\to(-5,-2)\) (matches \(P\)). So, \(\triangle DEF\) can be mapped to \(\triangle RPQ\) by a \(180^{\circ}\) rotation about the origin followed by a translation 2 units down.

Step3: Evaluate the Third Option

Since we found a rigid transformation (rotation + translation) that maps \(\triangle DEF\) to \(\triangle RPQ\), the third option is incorrect.

Step4: Re - evaluate the First Option

The first option says reflection across \(x\) - axis followed by translation 2 units left. After reflection across \(x\) - axis, \(D(1,3)\to(1,-3)\), \(F(1,0)\to(1,0)\), \(E(5,0)\to(5,0)\). Then translation 2 units left: \(D\to(-1,-3)\), \(F\to(-1,0)\), \(E\to(3,0)\). These coordinates don't match \(\triangle RPQ\)'s coordinates (\(R(-1,-5)\), \(Q(-1,-2)\), \(P(-5,-2)\)), so the first option is incorrect.

Answer:

Yes. \(\triangle DEF\) can be mapped to \(\triangle RPQ\) by a \(180^{\circ}\) rotation about the origin followed by a translation 2 units down.