A path between 3 cities is drawn on a map, creating the outlined region shown in the graph below.

Question

City A (2,2)
City B (4,8)
City C (8,4)

Which sequence of transformations would result in mapping the region onto itself?

a. a reflection across the v-axis and then a reflection across the x-axis
b. a rotation 90-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin
c. a rotation 90-degrees counterclockwise about the origin and then a rotation 180-degrees counterclockwise about the origin
d. a rotation 180-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin

Answer 1

To determine which sequence of transformations maps the region created by the cities A (2, 2), B (4, 8), and C (8, 4) onto itself, we can analyze each option step by step.

1. Option a: Reflection across the y-axis and then a reflection across the x-axis.

   • Reflecting across the y-axis transforms points \((x, y)\) to \((-x, y)\). In this case:
     • A (2, 2) becomes (-2, 2)
     • B (4, 8) becomes (-4, 8)
     • C (8, 4) becomes (-8, 4)
   • Now reflecting these points across the x-axis transforms them to \((-2, -2)\), \((-4, -8)\), and \((-8, -4)\).
   • This transformation does not preserve the original coordinates of the region.

2. Option b: A rotation 90-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin.

   • A 90-degree counterclockwise rotation transforms the points as follows:
     • A (2, 2) becomes (-2, 2)
     • B (4, 8) becomes (-8, 4)
     • C (8, 4) becomes (-4, 8)
   • A subsequent 270-degree counterclockwise rotation (which is equivalent to a 90-degree clockwise rotation) transforms those points:
     • (-2, 2) becomes (2, -2)
     • (-8, 4) becomes (4, 8)
     • (-4, 8) becomes (8, 4)
   • This sequence does not return the original points.

3. Option c: A rotation 90-degrees counterclockwise about the origin and then a rotation 180-degrees counterclockwise about the origin.

   • First, applying the 90-degree counterclockwise rotation:
     • A (2, 2) becomes (-2, 2)
     • B (4, 8) becomes (-8, 4)
     • C (8, 4) becomes (-4, 8)
   • Now, applying a 180-degree rotation gives:
     • (-2, 2) becomes (2, -2)
     • (-8, 4) becomes (8, -4)
     • (-4, 8) becomes (4, -8)
   • This does not map back onto the original points.

4. Option d: A rotation 180-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin.

   • The 180-degree rotation transforms:
     • A (2, 2) becomes (-2, -2)
     • B (4, 8) becomes (-4, -8)
     • C (8, 4) becomes (-8, -4)
   • A subsequent 270-degree counterclockwise rotation (or 90-degree clockwise rotation) gives us:
     • (-2, -2) becomes (-2, -2) (unchanged point is not a valid point in reflection)
     • (-4, -8) becomes (-8, -4)
     • (-8, -4) becomes (-4, 8)
   • This transformation does not return to the original placement either.

After analyzing all options, it appears that none of the transformations exactly map the original region back onto itself. However, visualizing the symmetry might reveal a missed reflection or rotation. It could be beneficial to check transformations visually or graphically using graph paper or software.

In conclusion, based on analysis, none of the provided options appear to yield the original positions, thus we need further verification to find the correct transformation that results in the map mapping onto itself.