Question

congruence transformations hw pt. 2 (show all work!) 1) figure a and its image after a transformation, figure a, are shown on the coordinate plane below. the two figures are congruent. which transformation has been performed to create the congruent figure a? a reflection over the x - axis b reflection over the y - axis c translation to the right of 9 units d clockwise rotation of 90° about the origin

Explanation:

Step1: Analyze reflection over x - axis

A reflection over the x - axis changes the sign of the y - coordinate of a point \((x,y)\) to \((x, - y)\). Let's take a vertex of Figure A, say if a vertex has coordinates \((x,y)\), after reflection over x - axis it would be \((x, - y)\). But looking at the figures, the vertical position change is not a reflection over x - axis (since the y - coordinates are not negated in a way that reflection over x - axis would do).

Step2: Analyze reflection over y - axis

A reflection over the y - axis changes the sign of the x - coordinate of a point \((x,y)\) to \((-x,y)\). This also does not match the transformation between Figure A and Figure A' as the horizontal flip (reflection over y - axis) is not what is happening here.

Step3: Analyze translation to the right

Translation to the right would keep the shape and orientation the same and just move the figure horizontally. But Figure A' has a different vertical and horizontal relationship in terms of orientation? Wait, no, actually, let's check the vertical shift. Wait, maybe we should check the rotation. Wait, no, let's check the reflection over x - axis? Wait, no, let's take a point from Figure A. Let's assume a vertex of Figure A is at \((2, - 5)\) (approximate from the graph) and a vertex of Figure A' is at \((2,5)\)? Wait, no, maybe I got the axes wrong. Wait, the x - axis is vertical? Wait, no, in the coordinate plane, usually x - axis is horizontal and y - axis is vertical, but in the given graph, the x - axis is vertical (since the arrow is vertical) and y - axis is horizontal (arrow is horizontal). So let's re - define: let's say the horizontal axis is y - axis (left - right) and vertical axis is x - axis (up - down). So a point \((x,y)\) where x is vertical (up - down) and y is horizontal (left - right). Then reflection over x - axis (vertical axis) would change the y - coordinate? Wait, no, maybe the graph has x - axis vertical (positive downwards) and y - axis horizontal (positive to the right). So a point \((x,y)\) with x increasing downwards and y increasing to the right. Now, Figure A is in the upper part (lower x - values) and Figure A' is in the lower part (higher x - values). A reflection over the x - axis (if x - axis is horizontal) no, wait, maybe the x - axis is the horizontal line (y = 0 in standard, but here maybe the middle horizontal line). Wait, let's take a vertex of Figure A: suppose one vertex is at (y = 5, x = - 6) (assuming y - axis horizontal, x - axis vertical, x positive down). Then Figure A' has a vertex at (y = 5, x = 6). Wait, no, that's reflection over y - axis? No, maybe I messed up. Wait, the correct way: in the given graph, the x - axis is vertical (the vertical line with arrow down) and y - axis is horizontal (horizontal line with arrow right). So a point is (x, y) where x is vertical (down is positive) and y is horizontal (right is positive). Now, Figure A is above (lower x) and Figure A' is below (higher x). A reflection over the x - axis (if x - axis is horizontal) no, wait, reflection over the horizontal line (let's say the line y = 0, but in this graph, the middle horizontal line). Wait, maybe the transformation is reflection over the x - axis. Wait, no, let's check the options again. Option A: reflection over x - axis. If we reflect a point \((x,y)\) over x - axis (in standard coordinate system, x horizontal, y vertical), the y - coordinate changes sign. But in this graph, if x is vertical (down positive) and y is horizontal (right positive), then reflecting over the x - axis (the horizontal line) would change the x - coordinate? No, I think I made a mistake in axis interpretation. Let's look at the figures: Figure A is in the upper half (above the middle horizontal line) and Figure A' is in the lower half (below the middle horizontal line). So the transformation is a reflection over the horizontal line (which is the x - axis in standard if we consider x horizontal, but here maybe the x - axis is horizontal). So a reflection over the x - axis (where x is horizontal) would flip the figure over the horizontal line, which matches the position of Figure A and Figure A'. Let's check the other options: translation to the right (C) would move the figure right, not up - down. Reflection over y - axis (B) would flip left - right, not up - down. Rotation (D) would change the orientation, but the figures have the same orientation (triangle pointing the same way), so it's a reflection over x - axis.

Answer:

A. reflection over the x - axis