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.)
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**: \( 40 \)
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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