Graph the triangle ABC after a 90-clockwise rotation around the center at the origin.
A(4,-3)
B(4,0)
C(8,0)
Xmin:
-10
Xmax:
10
Ymin:
-10
Ymax:
10
To graph the triangle ABC after a 90-clockwise rotation around the origin, we need to find the new coordinates of each point.
To rotate a point (x, y) 90 degrees clockwise around the origin, we can use the formula: (x', y') = (-y, x)
For point A(4, -3):
The new x-coordinate will be -(-3) = 3
The new y-coordinate will be 4
So, A' = (3, 4)
For point B(4, 0):
The new x-coordinate will be -(0) = 0
The new y-coordinate will be 4
So, B' = (0, 4)
For point C(8, 0):
The new x-coordinate will be -(0) = 0
The new y-coordinate will be 8
So, C' = (0, 8)
Now we can graph the triangle ABC and its rotated version:
Triangle ABC:
A(4, -3)
B(4, 0)
C(8, 0)
Triangle A'B'C':
A'(3, 4)
B'(0, 4)
C'(0, 8)
The graphed coordinates will correspond to the given Xmin, Xmax, Ymin, and Ymax values.
Xmin: -10
Xmax: 10
Ymin: -10
Ymax: 10
if AB =30 inches then find the distance of A′B after a reflection over the x-axis and a rotation of
To find the distance of A'B' after a reflection over the x-axis and a rotation, we need to find the new coordinates of points A and B, and then calculate the distance between the transformed points.
1. Reflection over the x-axis:
To reflect a point (x, y) over the x-axis, the new coordinates of the point will be (x, -y).
For point A(4, -3), after reflecting over the x-axis, the new coordinates will be A'(4, 3).
For point B(4, 0), after reflecting over the x-axis, the new coordinates will be B'(4, -0) which simplifies to B'(4, 0).
2. Rotation:
To rotate a point (x, y) 90 degrees clockwise around the origin, the new coordinates of the point can be found using the formula: (x', y') = (-y, x).
For point A'(4, 3), after rotating 90 degrees clockwise, the new coordinates will be A''(-3, 4).
For point B'(4, 0), after rotating 90 degrees clockwise, the new coordinates will be B''(0, 4).
3. Calculate the distance between A'' and B'':
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Using the formula, the distance between A''(-3, 4) and B''(0, 4) is:
d = sqrt((0 - (-3))^2 + (4 - 4)^2)
= sqrt(3^2 + 0^2)
= sqrt(9 + 0)
= sqrt(9)
= 3 inches.
Therefore, the distance between A'B' after a reflection over the x-axis and a rotation is 3 inches.
What are the coordinate points for A' if A(5, 11) after being translated right 2 and up 4.
To find the new coordinates for A' after being translated right 2 and up 4, we need to add 2 to the x-coordinate and add 4 to the y-coordinate of point A(5, 11).
The new x-coordinate for A' will be 5 + 2 = 7.
The new y-coordinate for A' will be 11 + 4 = 15.
Therefore, the coordinate points for A' after being translated right 2 and up 4 is (7, 15).
What is point A (8, 7) after a translation of 3 units left and 1 units down followed by a 180° clockwise rotation around the origin.
To find the new coordinates for point A (8, 7) after the given translation and rotation:
1. Translation:
To translate a point (x, y) a certain number of units left/right and up/down, we add/subtract the given values to the x and y coordinates respectively.
After translating 3 units left and 1 unit down, the new coordinates for point A are:
x-coordinate: 8 - 3 = 5
y-coordinate: 7 - 1 = 6
Therefore, after the translation, point A becomes A' with coordinates (5, 6).
2. Rotation:
To rotate a point (x, y) 180° clockwise around the origin, we use the formula (x', y') = (-x, -y).
Applying this formula to A' (5, 6), we get:
x-coordinate: -5
y-coordinate: -6
Therefore, after the 180° clockwise rotation, point A' becomes A'' with coordinates (-5, -6).
So, the final coordinates for point A after the translation and rotation are (-5, -6).
Find the dilation of 2 with center (0, 0) for the following points if A(−6, 1) B(8, 7)
A, ()
B′()
To find the dilation of 2 with center (0, 0) for points A(-6, 1) and B(8, 7), we need to multiply the x and y coordinates of each point by 2.
For point A(-6, 1):
The new x-coordinate will be -6 * 2 = -12.
The new y-coordinate will be 1 * 2 = 2.
So, A' = (-12, 2).
For point B(8, 7):
The new x-coordinate will be 8 * 2 = 16.
The new y-coordinate will be 7 * 2 = 14.
So, B' = (16, 14).
Therefore, the coordinates for point A' after the dilation are (-12, 2), and the coordinates for point B' after the dilation are (16, 14).