Question

Reflect triangle ABC
with respect to the x
-axis and the y
-axis. What are the vertices of triangle A′B′C′
?

(1 point)
Responses

The vertices of triangle A′B′C′
are A′(2,1)
, B′(6,3)
, and C′(1,3)
.
The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis 2 comma 1 right parenthesis , upper B prime left parenthesis 6 comma 3 right parenthesis , and upper C prime left parenthesis 1 comma 3 right parenthesis .

The vertices of triangle A′B′C′
are A′(−2,−1)
, B′(−6,−3)
, and C′(−1,−3)
.
The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis negative 2 comma negative 1 right parenthesis , upper B prime left parenthesis negative 6 comma negative 3 right parenthesis , and upper C prime left parenthesis negative 1 comma negative 3 right parenthesis .

The vertices of triangle A′B′C′
are A′(−2,1)
, B′(−6,3)
, and C′(−1,3)
.
The vertices of triangle upper A prime upper B prime upper C prime are upper A prime left parenthesis negative 2 comma 1 right parenthesis , upper B prime left parenthesis negative 6 comma 3 right parenthesis , and upper C prime left parenthesis negative 1 comma 3 right parenthesis .

The vertices of triangle A′B′C′
are A′(−2,1)
, B′(−1,3)
, and C′(−6,3)
.

Answer 1

To reflect triangle ABC across the x-axis and y-axis, let’s denote the original vertices of triangle A, B, and C as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).

### Step 1: Reflect Across the x-axis
When reflecting a point (x, y) across the x-axis, the new point becomes (x, -y). Thus, the reflected points of triangle ABC will be:

- \( A'(x_1, -y_1) \)
- \( B'(x_2, -y_2) \)
- \( C'(x_3, -y_3) \)

### Step 2: Reflect Across the y-axis
Next, reflecting the points from the previous step across the y-axis changes (x, y) to (-x, y). So we get:

- \( A''(-x_1, -y_1) \)
- \( B''(-x_2, -y_2) \)
- \( C''(-x_3, -y_3) \)

Combining these steps, the overall transformation results in the points:

- \( A''(-x_1, -y_1) \)
- \( B''(-x_2, -y_2) \)
- \( C''(-x_3, -y_3) \)

### Example
If triangle ABC has the vertices:
- \( A(2, 1) \)
- \( B(6, 3) \)
- \( C(1, 3) \)

Reflecting across the x-axis first:
- \( A'(2, -1) \)
- \( B'(6, -3) \)
- \( C'(1, -3) \)

Now reflecting those points across the y-axis:
- \( A''(-2, -1) \)
- \( B''(-6, -3) \)
- \( C''(-1, -3) \)

### Conclusion
The vertices of triangle A′B′C′ are:
- \( A'(-2, -1) \)
- \( B'(-6, -3) \)
- \( C'(-1, -3) \)

Thus, the correct response from the options provided would be:

**The vertices of triangle A′B′C′ are A′(−2,−1), B′(−6,−3), and C′(−1,−3).**