Question

square abcd is shown with four congruent images such that abcd ≅ fghi ≅ jklm ≅ nopq ≅ rstu. for which square is a reflection necessary to produce the transformed figure? jklm rstu fghi nopq

Explanation:

Step1: Analyze reflection concept

Reflection is a transformation that flips a figure over a line. We need to check each square's position relative to \(ABCD\).

Step2: Check \(FGHI\)

If we consider the \(y -\)axis (or \(x = 0\) line), the \(x -\)coordinates of corresponding vertices of \(ABCD\) (e.g., \(A(2, - 2)\), \(B(4,-2)\), \(C(4,-4)\), \(D(2,-4)\)) and \(FGHI\) (e.g., \(F(- 6,-2)\), \(G(-4,-2)\), \(H(-4,-4)\), \(I(-6,-4)\)) are symmetric about the \(y -\)axis. The formula for reflection over the \(y -\)axis is \((x,y)\to(-x,y)\). For a point \((x,y)\) in \(ABCD\), after reflection over \(y -\)axis, it becomes \((-x,y)\). For example, if \(A(2,-2)\), after reflection over \(y -\)axis, it is \((- 2,-2)\) (not \(FGHI\)’s vertex). But if we consider the \(x-\)axis (or \(y = 0\) line) reflection formula \((x,y)\to(x, - y)\), \(A(2,-2)\to(2,2)\) (not relevant). If we consider reflection over the vertical line \(x=-2\). The distance of a point \(x\) - coordinate of \(ABCD\) (e.g., \(x = 2\) for \(A\)) from \(x=-2\) is \(d=2-(-2)=4\). The corresponding \(x\) - coordinate of \(F\) is \(-2 - 4=-6\), \(G\): \(x=-2+(2 - 4)=-4\) (using the property of reflection over a vertical line \(x = a\): \(x\to2a - x\)).

Step3: Check other squares

• For \(JKLM\): It is a translation (movement without flipping) of \(ABCD\) (e.g., \(A(2,-2)\) to \(J(2,4)\) is a vertical shift).
• For \(NOPQ\): It is a translation (e.g., \(A(2,-2)\) to \(N(-4,4)\) involves both \(x\) and \(y\) - direction movement without flipping).
• For \(RSTU\): It is a combination of translation and no - reflection (compared to \(FGHI\) which is a reflection).

Answer:

FGHI