Question

performance task
for each part a through c, answer the question and include justifications.
jada works as an urban planner for a city. she is reviewing the plans for a shopping center, shown in the diagram. she is examining if the locations of the building can be changed to make better use of the land.
part a
jada wants to increase the size of a parking lot, so she wants to know if building 2 can be reflected across the y - axis without intersecting any of the other buildings in the diagram. is this possible? explain.
part b
can building 2 be rotated 90° clockwise around the origin without intersecting with any of the other buildings? explain.
part c
jada decides she wants to move building 1 so that its sides touch the sides of buildings 2 and 4, but do not overlap these buildings. what transformation can she perform? what are the new coordinates of building 1?
unit reflect
how can you use transformations to describe the congruence of two figures?

Explanation:

Step1: Recall reflection rule

The rule for reflecting a point $(x,y)$ across the $y -$axis is $(-x,y)$. Building 2 has vertices with positive $x -$coordinates. After reflection across the $y -$axis, its vertices will have negative $x -$coordinates. Looking at the diagram, there are no other buildings in the negative $x -$region where Building 2 will be reflected. So, reflection is possible.

Step2: Recall rotation rule

The rule for rotating a point $(x,y)$ 90° clockwise around the origin is $(y,-x)$. Analyze the position of Building 2. When rotated 90° clockwise, it will move into the space occupied by Building 1. So, a 90° clock - wise rotation around the origin is not possible without intersection.

Step3: Determine transformation for Building 1

To move Building 1 so its sides touch Buildings 2 and 4 without overlap, we can translate it. Let's assume the bottom - left vertex of Building 1 is currently at $(x_1,y_1)$ and we want to move it up and to the left. If we translate Building 1 2 units up and 2 units to the left. If the original vertices of Building 1 are, for example, $(x_1,y_1),(x_1 + a,y_1),(x_1,y_1 + b),(x_1 + a,y_1 + b)$ (where $a$ and $b$ are the length and width of the building), the new vertices will be $(x_1-2,y_1 + 2),(x_1 + a-2,y_1 + 2),(x_1-2,y_1 + b+2),(x_1 + a-2,y_1 + b+2)$.

Answer:

Part A: Yes, it is possible. When reflected across the y - axis, Building 2 will move to the negative x - region where there are no other buildings.
Part B: No, it is not possible. When rotated 90° clockwise around the origin, Building 2 will intersect with Building 1.
Part C: She can perform a translation. Translate Building 1 2 units up and 2 units to the left. The new coordinates depend on the original coordinates of Building 1's vertices. If the original bottom - left vertex is $(x,y)$, the new bottom - left vertex will be $(x - 2,y + 2)$ and the other vertices are adjusted accordingly based on the building's dimensions.