Question

Writing  Begin with point Aleft parenthesis 4 comma negative 6 right parenthesis. Reflections across the axes will produce points​ B, C, and D so that ABCD is a rectangle. What are the coordinates of point Upper B​? What is the length of side AUpper B​? Use pencil and paper. Explain how to use what you know about reflections to find the coordinates of the remaining 2 vertices. Explain how you can find the perimeter of the rectangle.
x
y
A
B
C
D



x y graph
Question content area bottom
Part 1
The coordinates of point Upper B is
  
enter your response here.
​(Type an ordered​ pair.)

Answers

Answered by GPT-4o mini
To create points B, C, and D after point A(4, -6) through reflections across the axes, we first need to find the coordinates of point B.

1. **Finding Point B**:
Point B is the reflection of point A across the y-axis. To reflect a point (x, y) across the y-axis, you negate the x-coordinate. Therefore, point B, which is the reflection of point A(4, -6), is:
- Coordinates of point B: \( B(-4, -6) \)

2. **Finding Length of Side AB**:
The length of side AB can be calculated using the distance formula between points A and B:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates of A and B:
- \( A(4, -6) \)
- \( B(-4, -6) \)

\[
\text{Distance AB} = \sqrt{((-4) - 4)^2 + ((-6) - (-6))^2}
\]
\[
= \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8
\]

3. **Finding Points C and D**:
- Point C will be the reflection of point B across the x-axis. To reflect a point (x, y) across the x-axis, you negate the y-coordinate. Therefore:
- Coordinates of point C: \( C(-4, 6) \)
- Point D will be the reflection of point A across the x-axis:
- Coordinates of point D: \( D(4, 6) \)

Now we have the vertices of the rectangle:
- A(4, -6)
- B(-4, -6)
- C(-4, 6)
- D(4, 6)

4. **Finding the Perimeter of Rectangle ABCD**:
The perimeter of a rectangle is calculated using the formula:
\[
\text{Perimeter} = 2(\text{length} + \text{width})
\]
In this case, the length of side AB is already found to be 8, and the height (or width) can be found using the distance between points A and C (or B and D), which is:
\[
\text{Distance AC} = \sqrt{((4) - (-4))^2 + ((-6) - (6))^2}
\]
\[
= \sqrt{(8)^2 + (-12)^2} = \sqrt{64 + 144} = \sqrt{208} = 4\sqrt{13}
\]

Now substituting the values into the perimeter formula:
\[
\text{Perimeter} = 2(8 + 12) = 2(20) = 40
\]

So, the final answers are:
- **Coordinates of point B**: \( B(-4, -6) \)
- **Length of side AB**: \( 8 \)
- **Perimeter of rectangle ABCD**: \( 40 \)
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