Question

consider figure a and figure b on the coordinate grid. part a to show that figure a and figure b are congruent, which series of transformations could be used? reflect figure a in the y - axis, followed by a reflection in the x - axis. translate figure a 3 units right, followed by a reflection in the x - axis. rotate figure a about the origin 180°, followed by a reflection in the x - axis. reflect figure a in the y - axis, followed by a 90° clockwise rotation about the origin. part b figure a is rotated 90° clockwise about the origin followed by a reflection in the x - axis, and then translated 1 unit up and 3 units to the left. use the drop - down menus to complete the sentence to describe a sequence that exhibits the congruence between the image of figure a and figure b. translate the image of figure a choose... unit(s) choose... and choose... unit(s) to the choose...

Explanation:

Step1: Analyze transformation rules

For a point \((x,y)\) in the coordinate - plane:

1. Reflection in the \(y\) - axis changes the point to \((-x,y)\).
2. Reflection in the \(x\) - axis changes the point to \((x, - y)\).
3. Translation \(a\) units right and \(b\) units up changes the point to \((x + a,y + b)\).
4. Rotation of \(180^{\circ}\) about the origin changes the point to \((-x,-y)\).
5. Rotation of \(90^{\circ}\) clockwise about the origin changes the point to \((y,-x)\).

Step2: Check Part A options

• Option 1: Reflect Figure \(A\) in the \(y\) - axis, then in the \(x\) - axis. Let a point \((x,y)\) on Figure \(A\). After reflection in the \(y\) - axis, it becomes \((-x,y)\), and after reflection in the \(x\) - axis, it becomes \((-x,-y)\).
• Option 2: Translate Figure \(A\) 3 units right (\((x,y)\to(x + 3,y)\)), then reflect in the \(x\) - axis (\((x + 3,y)\to(x + 3,-y)\)).
• Option 3: Rotate Figure \(A\) about the origin \(180^{\circ}\) (\((x,y)\to(-x,-y)\)), then reflect in the \(x\) - axis (\((-x,-y)\to(-x,y)\)).
• Option 4: Reflect Figure \(A\) in the \(y\) - axis (\((x,y)\to(-x,y)\)), then rotate \(90^{\circ}\) clockwise about the origin (\((-x,y)\to(y,x)\)).

By visual inspection of the figures on the grid, if we translate Figure \(A\) 3 units right and then reflect it in the \(x\) - axis, we can get Figure \(B\).

Step3: Analyze Part B

If Figure \(A\) is rotated \(90^{\circ}\) clockwise about the origin, reflected in the \(x\) - axis, and then translated 1 unit up and 3 units to the left.
The rotation \(90^{\circ}\) clockwise about the origin: \((x,y)\to(y,-x)\).
The reflection in the \(x\) - axis: \((y,-x)\to(y,x)\).
The translation 1 unit up and 3 units to the left: \((y,x)\to(y - 3,x + 1)\).
To get from the new - position of Figure \(A\) to Figure \(B\), we need to translate the image of Figure \(A\) 2 units right and 2 units down.

Answer:

Part A: Translate Figure A 3 units right, followed by a reflection in the x - axis.
Part B: Translate the image of Figure A 2 units right and 2 units down.