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.

Explanation:

Step1: Analyze translation

Let's assume a point \((x,y)\) on Figure 1. A translation of \(a\) units to the left and \(b\) units down changes the point to \((x - a,y - b)\). If we take a vertex of Figure 1, say \((5,3)\), a translation of 11 units to the left and 10 units down gives \((5-11,3 - 10)=(-6,-7)\) which can match a vertex of Figure 2. And a translation of 10 units down and 11 units to the left is the same transformation.

Step2: Analyze rotation

The rule for a counter - clockwise rotation of \(180^{\circ}\) about the origin for a point \((x,y)\) is \((-x,-y)\). For a point \((5,3)\) on Figure 1, after a \(180^{\circ}\) counter - clockwise rotation about the origin, we get \((-5,-3)\) which can match a vertex of Figure 2. The rule for a clockwise rotation of \(180^{\circ}\) about the origin for a point \((x,y)\) is also \((-x,-y)\).

Step3: Analyze reflection

A reflection over the \(x\) - axis changes \((x,y)\) to \((x,-y)\), and then a reflection over the \(y\) - axis changes \((x,-y)\) to \((-x,-y)\) which is equivalent to a \(180^{\circ}\) rotation about the origin.

Answer:

A. A translation of 11 units to the left and 10 units below.
B. A counterclockwise rotation by \(180^{\circ}\) 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 \(180^{\circ}\) about the origin.