Question
Graph the triangle ABC after a 90-clockwise rotation around the center at the origin.
Answers
Answer
if ′A′B′ 30 inches then find the distance of ′A′B′ after a reflection over the x-axis and a rotation of 341° clockwise.
Answered by
GPT 3.5
To graph the triangle ABC after a 90-clockwise rotation around the origin, we need to rotate each of its three vertices 90 degrees clockwise around the origin.
Let's say the coordinates of point A are (x1, y1), the coordinates of point B are (x2, y2), and the coordinates of point C are (x3, y3).
To rotate a point (x, y) 90 degrees clockwise around the origin, we can use the following formulas:
x' = y
y' = -x
Applying these formulas to each vertex of the triangle, we get:
A' = (y1, -x1)
B' = (y2, -x2)
C' = (y3, -x3)
Thus, the coordinates of the rotated triangle ABC are:
A' = (y1, -x1)
B' = (y2, -x2)
C' = (y3, -x3)
To visualize the rotation, you can plot the original triangle ABC and then plot the rotated triangle ABC' on the same coordinate plane.
Let's say the coordinates of point A are (x1, y1), the coordinates of point B are (x2, y2), and the coordinates of point C are (x3, y3).
To rotate a point (x, y) 90 degrees clockwise around the origin, we can use the following formulas:
x' = y
y' = -x
Applying these formulas to each vertex of the triangle, we get:
A' = (y1, -x1)
B' = (y2, -x2)
C' = (y3, -x3)
Thus, the coordinates of the rotated triangle ABC are:
A' = (y1, -x1)
B' = (y2, -x2)
C' = (y3, -x3)
To visualize the rotation, you can plot the original triangle ABC and then plot the rotated triangle ABC' on the same coordinate plane.
Answered by
GPT 3.5
To find the distance of A'B' after a reflection over the x-axis and a rotation of 341° clockwise, we need to perform the two transformations on the segment AB and then measure the length of the resulting segment.
1. Reflection over the x-axis:
A' = (x1, -y1)
B' = (x2, -y2)
2. Rotation of 341° clockwise:
To rotate a point (x, y) 341° clockwise around the origin, we can use the following formulas:
x' = x * cos(theta) + y * sin(theta)
y' = -x * sin(theta) + y * cos(theta)
Here, theta = 341° or 341 * ( π/180 ) radians.
Applying these formulas to each endpoint of the segment A'B', we get:
A'' = (A'x * cos(theta) + A'y * sin(theta), -A'x * sin(theta) + A'y * cos(theta))
B'' = (B'x * cos(theta) + B'y * sin(theta), -B'x * sin(theta) + B'y * cos(theta))
Finally, we can calculate the distance between A'' and B'' to find the length of A'B''.
A'B'' = sqrt((A''x - B''x)^2 + (A''y - B''y)^2)
Make sure to substitute the appropriate coordinates into the formulas to find the final distance.
1. Reflection over the x-axis:
A' = (x1, -y1)
B' = (x2, -y2)
2. Rotation of 341° clockwise:
To rotate a point (x, y) 341° clockwise around the origin, we can use the following formulas:
x' = x * cos(theta) + y * sin(theta)
y' = -x * sin(theta) + y * cos(theta)
Here, theta = 341° or 341 * ( π/180 ) radians.
Applying these formulas to each endpoint of the segment A'B', we get:
A'' = (A'x * cos(theta) + A'y * sin(theta), -A'x * sin(theta) + A'y * cos(theta))
B'' = (B'x * cos(theta) + B'y * sin(theta), -B'x * sin(theta) + B'y * cos(theta))
Finally, we can calculate the distance between A'' and B'' to find the length of A'B''.
A'B'' = sqrt((A''x - B''x)^2 + (A''y - B''y)^2)
Make sure to substitute the appropriate coordinates into the formulas to find the final distance.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.