Question

which of the following transformations would preserve distance when applied to the pre - image? select all that apply. reflection over the y - axis dilation centered at the origin with a scale factor of 2 rotation 180° counterclockwise about the origin dilation centered at the origin with a scale factor of 0.5 reflection over the line ( y=-x ) translation along the vector ( <1, - 3> )

Explanation:

• Reflection over the \(y\)-axis: A reflection is a rigid transformation. Rigid transformations preserve the shape and size of the pre - image, which means they preserve distance. The formula for reflecting a point \((x,y)\) over the \(y\) - axis is \((-x,y)\). The distance between two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and for their images \(P'(-x_1,y_1)\) and \(Q'(-x_2,y_2)\), the distance \(d'=\sqrt{(-x_2+x_1)^2+(y_2 - y_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=d\).
• Rotation \(180^{\circ}\) counter - clockwise about the origin: A rotation is a rigid transformation. The formula for rotating a point \((x,y)\) \(180^{\circ}\) counter - clockwise about the origin is \((-x,-y)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) be two points. The distance between \(P\) and \(Q\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and for their images \(P'(-x_1,-y_1)\) and \(Q'(-x_2,-y_2)\), the distance \(d'=\sqrt{(-x_2 + x_1)^2+(-y_2 + y_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=d\).
• Reflection over the line \(y=-x\): A reflection is a rigid transformation. The formula for reflecting a point \((x,y)\) over the line \(y =-x\) is \((-y,-x)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) be two points. The distance between \(P\) and \(Q\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and for their images \(P'(-y_1,-x_1)\) and \(Q'(-y_2,-x_2)\), the distance \(d'=\sqrt{(-y_2 + y_1)^2+(-x_2 + x_1)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=d\).
• **Translation along the vector \(\langle1,-3

angle\)**: A translation is a rigid transformation. The formula for translating a point \((x,y)\) along the vector \(\langle a,b
angle\) is \((x + a,y + b)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) be two points. Their images are \(P'(x_1 + 1,y_1-3)\) and \(Q'(x_2 + 1,y_2-3)\). The distance between \(P'\) and \(Q'\) is \(d'=\sqrt{(x_2 + 1-(x_1 + 1))^2+(y_2-3-(y_1 - 3))^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=d\).

• Dilation centered at the origin with a scale factor of \(2\): A dilation is a non - rigid transformation when the scale factor \(k

eq1\). The formula for dilating a point \((x,y)\) centered at the origin with a scale factor \(k\) is \((kx,ky)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) be two points. The distance between \(P\) and \(Q\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and for their images \(P'(2x_1,2y_1)\) and \(Q'(2x_2,2y_2)\), the distance \(d'=\sqrt{(2x_2-2x_1)^2+(2y_2 - 2y_1)^2}=2\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=2d\).

• Dilation centered at the origin with a scale factor of \(0.5\): The formula for dilating a point \((x,y)\) centered at the origin with a scale factor \(k = 0.5\) is \((0.5x,0.5y)\). Let \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) be two points. The distance between \(P\) and \(Q\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and for their images \(P'(0.5x_1,0.5y_1)\) and \(Q'(0.5x_2,0.5y_2)\), the distance \(d'=\sqrt{(0.5x_2-0.5x_1)^2+(0.5y_2 - 0.5y_1)^2}=0.5\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=0.5d\).

Answer:

• Reflection over the \(y\) - axis
• Rotation \(180^{\circ}\) counterclockwise about the origin
• Reflection over the line \(y=-x\)
• Translation along the vector \(\langle1,-3

angle\)