Question

1. select all of the following compositions that preserve distance.

a a reflection over the y - axis followed by a dilation with a scale factor of 0.5.
b a translation (x,y)→(x - 1,y + 2) followed by a reflection over the y - axis.
c a dilation with a scale factor of 1.25 followed by a 180 degree rotation about the origin.
d a rotation 90 degrees clockwise about the origin followed by a reflection over the x - axis.
e a reflection over the x - axis followed by a rotation 90 degrees clockwise about the origin.
f a 90 degree counterclockwise rotation about the origin followed by the translation (x,y)→(x - 6,y - 2).

Explanation:

Step1: Understand distance - preserving transformations

Isometries (distance - preserving transformations) include translations, reflections, and rotations. Dilations are not isometries (unless the scale factor \(k = 1\)).

• Translation: A translation \((x,y)\to(x - h,y - k)\) (in this case, for option B \((x,y)\to(x - 1,y + 2)\) is a translation. Translations preserve distances. If we have two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\), the distance \(d(P,Q)=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). After translation \(P'(x_1 - 1,y_1+2)\) and \(Q'(x_2 - 1,y_2 + 2)\), \(d(P',Q')=\sqrt{((x_2 - 1)-(x_1 - 1))^2+((y_2 + 2)-(y_1 + 2))^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Reflection: A reflection over an axis (e.g., for option A, reflection over the \(y\) - axis: \((x,y)\to(-x,y)\)). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\). After reflection over the \(y\) - axis \(P'(-x_1,y_1)\) and \(Q'(-x_2,y_2)\). Then \(d(P',Q')=\sqrt{(-x_2+ x_1)^2+(y_2 - y_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Rotation: A rotation about the origin (e.g., for option D, a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y,-x)\)). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\). After rotation \(P'(y_1,-x_1)\) and \(Q'(y_2,-x_2)\). Then \(d(P',Q')=\sqrt{(y_2 - y_1)^2+(-x_2 + x_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For option E, a reflection over the \(x\) - axis \((x,y)\to(x,-y)\) followed by a \(90^{\circ}\) clockwise rotation about the origin \((x,-y)\to(-y,-x)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\). After the first transformation \(P_1(x_1,-y_1)\) and \(Q_1(x_2,-y_2)\), after the second transformation \(P_2(-y_1,-x_1)\) and \(Q_2(-y_2,-x_2)\). \(d(P_2,Q_2)=\sqrt{(-y_2 + y_1)^2+(-x_2 + x_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For option F, a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\) followed by a translation \((-y,x)\to(-y - 6,x - 2)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\). After rotation \(P_1(-y_1,x_1)\) and \(Q_1(-y_2,x_2)\), after translation \(P_2(-y_1-6,x_1 - 2)\) and \(Q_2(-y_2-6,x_2 - 2)\). \(d(P_2,Q_2)=\sqrt{(-y_2 + y_1)^2+(x_2 - x_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Dilation: For option C, a dilation with a scale factor \(k = 1.25\). If \(P(x_1,y_1)\) and \(Q(x_2,y_2)\), after dilation \(P'(1.25x_1,1.25y_1)\) and \(Q'(1.25x_2,1.25y_2)\). Then \(d(P',Q')=\sqrt{(1.25x_2-1.25x_1)^2+(1.25y_2 - 1.25y_1)^2}=1.25\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}

eq\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (since \(1.25
eq1\)).

Answer:

A. A reflection over the \(y\) - axis followed by a dilation with a scale factor of \(0.5\) (reflection is an isometry, but if we consider the composition, the dilation part: if we first reflect \((x,y)\to(-x,y)\) and then dilate \((-x,y)\to(-0.5x,0.5y)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\). After reflection \(P_1(-x_1,y_1)\) and \(Q_1(-x_2,y_2)\), after dilation \(P_2(-0.5x_1,0.5y_1)\) and \(Q_2(-0.5x_2,0.5y_2)\). \(d(P_2,Q_2)=\sqrt{(-0.5x_2 + 0.5x_1)^2+(0.5y_2 - 0.5y_1)^2}=0.5\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}
eq\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (incorrect).
B. A translation \((x,y)\to(x - 1,y + 2)\) (correct, as shown in the distance - preservation proof above).
D. A rotation \(90\) degrees clockwise about the origin followed by a reflection over the \(x\) - axis (correct, as shown in the distance - preservation proof above).
E. A reflection over the \(x\) - axis followed by a rotation \(90\) degrees clockwise about the origin (correct, as shown in the distance - preservation proof above).
F. A \(90\) degree counterclockwise rotation about the origin followed by the translation \((x,y)\to(x - 6,y - 2)\) (correct, as shown in the distance - preservation proof above).

So the correct answers are B, D, E, F.