Rectangle ABCD

Question

has vertices A(−1,−2)
, B(−1,−8)
, C(−8,−8)
, and D(−8,−2)
.
The rectangle ABCD
is reflected on the x
-axis and the y
-axis. Find the coordinates of the vertices for rectangle A′B′C′D′
.(1 point)
Responses

Rectangle A′B′C′D′
has vertices A′(1,−2)
, B′(1,−8)
, C′(8,−8)
, and D′(8,−2)
.
Rectangle upper A prime upper B prime upper C prime upper D prime has vertices upper A prime left parenthesis 1 comma negative 2 right parenthesis , upper B prime left parenthesis 1 comma negative 8 right parenthesis , upper C prime left parenthesis 8 comma negative 8 right parenthesis , and upper D prime left parenthesis 8 comma negative 2 right parenthesis .

Rectangle A′B′C′D′
has vertices A′(1,2)
, B′(1,8)
, C′(8,8)
, and D′(8,2)
.
Rectangle upper A prime upper B prime upper C prime upper D prime has vertices upper A prime left parenthesis 1 comma 2 right parenthesis , upper B prime left parenthesis 1 comma 8 right parenthesis , upper C prime left parenthesis 8 comma 8 right parenthesis , and upper D prime left parenthesis 8 comma 2 right parenthesis .

Rectangle A′B′C′D′
has vertices A′(1,2)
, B′(1,8)
, and C′(8,8)
.
Rectangle upper A prime upper B prime upper C prime upper D prime has vertices upper A prime left parenthesis 1 comma 2 right parenthesis , upper B prime left parenthesis 1 comma 8 right parenthesis , and upper C prime left parenthesis 8 comma 8 right parenthesis .

Rectangle A′B′C′D′
has vertices A′(−1,2)
, B′(−1,8)
, C′(−8,8)
, and D′(−8,2)
.

Answer 1

To find the coordinates of the vertices of rectangle \( A'B'C'D' \) after reflecting rectangle \( ABCD \) over the x-axis and then the y-axis, we will follow these steps:

1. Reflect over the x-axis: This changes the y-coordinates of each vertex to their opposite value (i.e., \( (x, y) \) becomes \( (x, -y) \)).
2. Reflect over the y-axis: This changes the x-coordinates of each vertex to their opposite value (i.e., \( (x, y) \) becomes \( (-x, y) \)).

Starting with the vertices of rectangle \( ABCD \):

• \( A(-1, -2) \)
• \( B(-1, -8) \)
• \( C(-8, -8) \)
• \( D(-8, -2) \)

Step 1: Reflect over the x-axis

• \( A'(-1, -(-2)) = (-1, 2) \)
• \( B'(-1, -(-8)) = (-1, 8) \)
• \( C'(-8, -(-8)) = (-8, 8) \)
• \( D'(-8, -(-2)) = (-8, 2) \)

After this reflection, the coordinates become:

• \( A'(-1, 2) \)
• \( B'(-1, 8) \)
• \( C'(-8, 8) \)
• \( D'(-8, 2) \)

Step 2: Reflect over the y-axis

• \( A''(-(-1), 2) = (1, 2) \)
• \( B''(-(-1), 8) = (1, 8) \)
• \( C''(-(-8), 8) = (8, 8) \)
• \( D''(-(-8), 2) = (8, 2) \)

After this reflection, the final coordinates become:

• \( A''(1, 2) \)
• \( B''(1, 8) \)
• \( C''(8, 8) \)
• \( D''(8, 2) \)

Thus, the coordinates of the vertices for rectangle \( A'B'C'D' \) are:

• \( A'(1, 2) \)
• \( B'(1, 8) \)
• \( C'(8, 8) \)
• \( D'(8, 2) \)

The correct response based on the reflections is: Rectangle A′B′C′D′ has vertices A′(1, 2), B′(1, 8), C′(8, 8), and D′(8, 2).