Question

select all the transformations that map figure 1 onto figure 2. a a translation of 11 units to the left and 10 units below. b a counterclockwise rotation by 180° about the origin. c a translation of 10 units below and 11 units to the left. d a reflection over the x - axis and then over the y - axis. e a clockwise rotation by 180° about the origin. f a reflection over the y - axis and then over the x - axis.

Explanation:

Step1: Analyze translation

Translation involves moving the figure without rotation or reflection. A translation of 11 units to the left and 10 units below (or equivalently 10 units below and 11 units to the left) can map one - figure to another if their shapes and orientations are the same. Options A and C describe the same translation.

Step2: Analyze rotation

A 180 - degree rotation (clockwise or counter - clockwise) about the origin changes the orientation of the figure. For a 180 - degree rotation about the origin, the coordinates \((x,y)\) of a point on the original figure are transformed to \((-x,-y)\). Options B and E are equivalent as a 180 - degree counter - clockwise rotation is the same as a 180 - degree clockwise rotation in terms of the final position of the figure.

Step3: Analyze reflection

A reflection over the \(x\) - axis changes the sign of the \(y\) - coordinate (\((x,y)\to(x, - y)\)) and a reflection over the \(y\) - axis changes the sign of the \(x\) - coordinate (\((x,y)\to(-x,y)\)). A reflection over the \(x\) - axis and then over the \(y\) - axis (\((x,y)\to(-x,-y)\)) is equivalent to a 180 - degree rotation about the origin. Options D and F are equivalent to a 180 - degree rotation about the origin.

Answer:

A. A translation of 11 units to the left and 10 units below.
B. A counterclockwise rotation by 180° about the origin.
C. A translation of 10 units below and 11 units to the left.
D. A reflection over the x - axis and then over the y - axis.
E. A clockwise rotation by 180° about the origin.
F. A reflection over the y - axis and then over the x - axis.