What is point A (-7, -9) after a translation of 5 units right and 3 units up followed by a reflection over the x-axis
To find the position of point A after a translation of 5 units right and 3 units up, we add 5 to the x-coordinate and add 3 to the y-coordinate.
After the translation, point A becomes (-7 + 5, -9 + 3) = (-2, -6).
To reflect point A over the x-axis, we keep the x-coordinate the same and negate the y-coordinate.
After the reflection, point A becomes (-2, -(-6)) = (-2, 6).
Therefore, point A is located at (-2, 6) after a translation of 5 units right and 3 units up followed by a reflection over the x-axis.
If = AB‾ = 27 inches then find the distance of ' ‾ A ′ B ′ after a reflection over the x-axis and a rotation of ° 357°
To find the distance of A' B' after the reflection over the x-axis and a rotation of 357°, we need to perform each transformation separately and then find the distance between the two resulting points.
1. Reflection Over the X-Axis:
When we reflect a point over the x-axis, the y-coordinate becomes its negative counterpart while the x-coordinate remains the same.
So, A' (reflected over the x-axis) becomes (-x-coordinate of A, -y-coordinate of A) = (-x-coordinate of A, -(-9)) = (-7, 9).
B' (reflected over the x-axis) becomes (-x-coordinate of B, -y-coordinate of B) = (-x-coordinate of B, -(-9)) = (-7, 9).
2. Rotation of 357°:
To perform a rotation of 357° counterclockwise, we can use the following transformation matrix:
| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
The point A' (after reflection) = (-7, 9) becomes (x1', y1') = (-7, 9).
The point B' (after reflection) = (-7, 9) becomes (x2', y2') = (-7, 9).
Using the transformation matrix, we get:
| x1‾′ ‾ y1‾′ | = | cos(357°) -sin(357°) | * | x1' y1' |
| x2‾′ ‾ y2‾′ | | sin(357°) cos(357°) | | x2' y2' |
Calculating the matrix products, we find:
x1‾′ = cos(357°) * x1' - sin(357°) * y1' = cos(357°) * (-7) - sin(357°) * 9,
y1‾′ = sin(357°) * x1' + cos(357°) * y1' = sin(357°) * (-7) + cos(357°) * 9,
x2‾′ = cos(357°) * x2' - sin(357°) * y2' = cos(357°) * (-7) - sin(357°) * 9,
y2‾′ = sin(357°) * x2' + cos(357°) * y2' = sin(357°) * (-7) + cos(357°) * 9.
Finally, we can find the distance between A' B' by using the distance formula:
Distance = √((x2‾′ - x1‾′)^2 + (y2‾′ - y1‾′)^2).
Plugging in the values we found above, we can calculate the distance.
= inches
I apologize for the confusion, but without the specific value for the angle of rotation, I am unable to provide an exact numerical answer for the distance between A' and B' after the reflection over the x-axis and a rotation. If you have a specific value for the rotation angle, please provide it, and I would be happy to assist you further.
if AB=27 inches then find the distance of A'B' after a reflection over the x-axis and a rotation of 357° clockwise
To find the distance of A'B' after a reflection over the x-axis and a rotation of 357° clockwise, we need to first perform the reflection and then the rotation.
1. Reflection Over the X-Axis:
When we reflect a point over the x-axis, the y-coordinate becomes its negative counterpart while the x-coordinate remains the same.
So, point A' (reflected over the x-axis) becomes (-7, -(-9)) = (-7, 9), and point B' (reflected over the x-axis) becomes (-7, -(-9)) = (-7, 9).
2. Rotation of 357° Clockwise:
To perform a rotation of 357° clockwise, we need to rotate the points A' and B' about the origin (0, 0).
Using the rotation matrix:
| cos(θ) sin(θ) |
| -sin(θ) cos(θ) |
We have:
x1' = -7
y1' = 9
x2' = -7
y2' = 9
Plugging the above values and the rotation angle of 357° into the rotation matrix, we get:
x1'' = cos(357°) * x1' + sin(357°) * y1'
y1'' = -sin(357°) * x1' + cos(357°) * y1'
x2'' = cos(357°) * x2' + sin(357°) * y2'
y2'' = -sin(357°) * x2' + cos(357°) * y2'
After calculating these values, we can use the distance formula to find the distance between A'B':
Distance = √((x2'' - x1'')^2 + (y2'' - y1'')^2)
Substituting the values we found, we can calculate the exact distance between A'B'.
What are the coordinate points of A'(4,4)if A undergoes a rotation of:
90° clockwise =
180° clockwise =
270° clockwise =
To find the coordinate points of A' after A undergoes rotations, we can use the rotational transformation formulas. Let's calculate A' for each rotation specified:
1. 90° Clockwise Rotation:
The 90° clockwise rotation transforms the point (x, y) to (-y, x). Applying this formula to point A (4, 4):
A' = (-4, 4).
2. 180° Clockwise Rotation:
The 180° clockwise rotation transforms the point (x, y) to (-x, -y). Applying this formula to point A (4, 4):
A' = (-4, -4).
3. 270° Clockwise Rotation:
The 270° clockwise rotation transforms the point (x, y) to (y, -x). Applying this formula to point A (4, 4):
A' = (4, -4).
So, when A undergoes:
- 90° clockwise rotation, A' is located at (-4, 4).
- 180° clockwise rotation, A' is located at (-4, -4).
- 270° clockwise rotation, A' is located at (4, -4).