If AB=11 inches then find the distance of A′ B′ after a reflection over the x-axis and a rotation of 162° clockwise.
=inches
To find the distance of A' B' after a reflection over the x-axis and a rotation of 162° clockwise, we first need to find the coordinates of A' and B' after each transformation.
Let's assume the original coordinates of A are (xA, yA), and the original coordinates of B are (xB, yB).
Reflection over the x-axis:
The reflection over the x-axis changes the sign of the y-coordinate, while the x-coordinate remains the same.
So, the coordinates of A' after reflection over the x-axis are (xA, -yA).
Similarly, the coordinates of B' after reflection over the x-axis are (xB, -yB).
Rotation of 162° clockwise:
To rotate a point (x, y) clockwise by an angle θ, we can use the following formulas:
x' = x * cos(θ) + y * sin(θ)
y' = -x * sin(θ) + y * cos(θ)
Let's apply these formulas to the coordinates of A' and B' after reflection over the x-axis:
Coordinates of A' after reflection over the x-axis: (xA, -yA)
xA' = xA * cos(162°) + (-yA) * sin(162°)
yA' = -xA * sin(162°) + (-yA) * cos(162°)
Coordinates of B' after reflection over the x-axis: (xB, -yB)
xB' = xB * cos(162°) + (-yB) * sin(162°)
yB' = -xB * sin(162°) + (-yB) * cos(162°)
The distance between A' and B' can be found using the distance formula:
d = √((xB' - xA')^2 + (yB' - yA')^2)
Now, let's calculate these values.
First, let's assume the coordinates of A are (2, 3) and the coordinates of B are (5, 4).
Reflection over the x-axis:
Coordinates of A' after reflection over the x-axis: (2, -3)
Coordinates of B' after reflection over the x-axis: (5, -4)
Rotation of 162° clockwise:
Coordinates of A' after reflection over the x-axis and rotation of 162° clockwise: (xA', yA')
xA' = 2 * cos(162°) + (-(-3)) * sin(162°)
yA' = -2 * sin(162°) + (-(-3)) * cos(162°)
Coordinates of B' after reflection over the x-axis and rotation of 162° clockwise: (xB', yB')
xB' = 5 * cos(162°) + (-(-4)) * sin(162°)
yB' = -5 * sin(162°) + (-(-4)) * cos(162°)
Finally, we can calculate the distance d using the distance formula:
d = √((xB' - xA')^2 + (yB' - yA')^2)
Rotate triangle ABC 90° clockwise around the center of the graph and then reflect over the y-axis. Plot triangle DEF.
To rotate triangle ABC 90° clockwise around the center of the graph, we need to find the new coordinates of each vertex.
Let's assume the original coordinates of vertex A are (xA, yA), of vertex B are (xB, yB), and of vertex C are (xC, yC).
To rotate a point (x, y) 90° clockwise, the new coordinates can be found using the following formulas:
x' = y
y' = -x
Applying these formulas to each vertex, we get:
Coordinates of vertex D after rotation: (yA, -xA)
Coordinates of vertex E after rotation: (yB, -xB)
Coordinates of vertex F after rotation: (yC, -xC)
Next, we need to reflect triangle DEF over the y-axis. To do this, we simply change the sign of the x-coordinate while keeping the y-coordinate the same.
So, the final coordinates of each vertex after both transformations are:
Coordinates of vertex D after rotation and reflection: (-yA, -xA)
Coordinates of vertex E after rotation and reflection: (-yB, -xB)
Coordinates of vertex F after rotation and reflection: (-yC, -xC)
Plotting these new coordinates, we can now locate triangle DEF on the graph.
Rotate triangle ABC 90° clockwise around the center of the graph and then reflect over the y-axis. Plot triangle DEF.
A(2,3)
C(4,3)
B(2,6)
Xmin:
-10
Xmax:
10
Ymin:
-10
Ymax:
10
To rotate triangle ABC 90° clockwise around the center of the graph and then reflect over the y-axis, we first need to find the coordinates of each vertex after each transformation.
Let's assume the original coordinates of vertex A are (2, 3), of vertex B are (2, 6), and of vertex C are (4, 3).
Rotation 90° clockwise:
To rotate a point (x, y) 90° clockwise, the new coordinates can be found using the following formulas:
x' = y
y' = -x
Referring to triangle ABC, we can apply these formulas to each vertex to calculate the coordinates after the rotation:
Coordinates of vertex D after rotation: (3, -2)
Coordinates of vertex E after rotation: (6, -2)
Coordinates of vertex F after rotation: (3, -4)
Now that we have the new coordinates after the rotation, we can proceed to reflect triangle DEF over the y-axis. To do this, we change the sign of the x-coordinate while keeping the y-coordinate the same.
So, the final coordinates of each vertex after the rotation and reflection are:
Coordinates of vertex D after rotation and reflection: (-3, -2)
Coordinates of vertex E after rotation and reflection: (-6, -2)
Coordinates of vertex F after rotation and reflection: (-3, -4)
Now, let's plot these coordinates on the graph with the given limits:
Xmin: -10
Xmax: 10
Ymin: -10
Ymax: 10
The coordinates after the rotation and reflection are:
D: (-3, -2)
E: (-6, -2)
F: (-3, -4)
Plotting these points on the graph, we get triangle DEF in its final position.
Reflect triangle ABC over the y-axis and then rotate triangle 90° clockwise around the center. Plot triangle DEF.
A(2,3)
C(4,3)
B(2,6)
Xmin:
-10
Xmax:
10
Ymin:
-10
Ymax:
10
To reflect triangle ABC over the y-axis and then rotate it 90° clockwise around the center, we need to find the coordinates of each vertex after each transformation.
Let's assume the original coordinates of vertex A are (2, 3), of vertex B are (2, 6), and of vertex C are (4, 3).
Reflecting over the y-axis:
To reflect a point (x, y) over the y-axis, we change the sign of the x-coordinate while keeping the y-coordinate the same.
Applying this transformation to each vertex, we get the new coordinates:
Coordinates of vertex D after reflection: (-2, 3)
Coordinates of vertex E after reflection: (-2, 6)
Coordinates of vertex F after reflection: (-4, 3)
Next, we rotate triangle DEF 90° clockwise around the center.
Rotation 90° clockwise:
To rotate a point (x, y) 90° clockwise, the new coordinates can be found using the following formulas:
x' = y
y' = -x
Applying this rotation to each vertex, we get the final coordinates:
Coordinates of vertex D after reflection and rotation: (3, -2)
Coordinates of vertex E after reflection and rotation: (6, -2)
Coordinates of vertex F after reflection and rotation: (3, -4)
Now we can plot triangle DEF on the graph with the given limits:
Xmin: -10
Xmax: 10
Ymin: -10
Ymax: 10
The final coordinates of triangle DEF after the reflection and rotation are:
D: (3, -2)
E: (6, -2)
F: (3, -4)
Plotting these points on the graph, we can see triangle DEF in its final position.
Translate triangle ABC left 3, down 5 and then perform a dilation of 4 around the center at the origin.
A(2,3)
C(4,3)
B(2,6)
Xmin:
-10
Xmax:
10
Ymin:
-10
Ymax:
10
To translate triangle ABC left 3 units and down 5 units, we subtract 3 from the x-coordinates and subtract 5 from the y-coordinates of each vertex.
Let's assume the original coordinates of vertex A are (2, 3), of vertex B are (2, 6), and of vertex C are (4, 3).
After the translation, the new coordinates will be:
A' = (2 - 3, 3 - 5) = (-1, -2)
B' = (2 - 3, 6 - 5) = (-1, 1)
C' = (4 - 3, 3 - 5) = (1, -2)
Next, we will perform a dilation of 4 around the origin. To dilate a point (x, y) by a scale factor of 4, we multiply each coordinate by 4.
Applying the dilation to each vertex, the new coordinates will be:
A'' = (-1 * 4, -2 * 4) = (-4, -8)
B'' = (-1 * 4, 1 * 4) = (-4, 4)
C'' = (1 * 4, -2 * 4) = (4, -8)
Now, let's plot triangle ABC on the graph with the given limits:
Xmin: -10
Xmax: 10
Ymin: -10
Ymax: 10
The final coordinates of triangle ABC after the translation and dilation are:
A'': (-4, -8)
B'': (-4, 4)
C'': (4, -8)
Plotting these points on the graph, we can see triangle A''B''C'' in its final position.
Consider the point (-2, -1). We must move this point to (10, 10). We can use a dilation, rotation, reflection and translation.
Create a series of rules to take point (-2, -1) to (10, 10). You must use at least 3 of the 4 different types to earn full-credit.
A rule that uses one distinct type of rule = 5 points.
A rule that uses two distinct types of rules = 15 points