If AB¯¯¯¯¯¯¯¯=
27 inches, then find the distance of A′B′¯¯ after a reflection over the x-axis and a translation of (x + 6, y − 2)
A′B′¯¯¯¯¯¯¯¯¯¯=
The distance between two points can be found using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
Given AB = 27 inches, we can write the equation as:
27 = √((x2 - x1)^2 + (y2 - y1)^2)
After reflecting over the x-axis, the y-coordinate changes sign. So, if A is (x1, y1), then A' will be (x1, -y1). Similarly, if B is (x2, y2), then B' will be (x2, -y2).
We can write the new distance equation as:
A'B' = √((x2 - x1)^2 + (-y2 + y1)^2)
After the translation of (x + 6, y - 2), the new coordinates for A' and B' will be (x1 + 6, -y1 - 2) and (x2 + 6, -y2 - 2), respectively.
Substituting these coordinates into the distance equation, we get:
A'B' = √(((x2 + 6) - (x1 + 6))^2 + ((-y2 - 2) - (-y1 - 2))^2)
= √((x2 - x1)^2 + (-y2 + y1)^2)
= √((x2 - x1)^2 + (y2 - y1)^2)
Since A'B' is equal to AB, the distance remains unchanged, so
A'B' = 27 inches.
What is the final position of point A(7, 8) after a translation of 2 units left, 1 unit down, and followed by a 180° clockwise rotation around the origin?
A'=
To find the final position of point A after the given transformations, we can follow these steps:
1. Translation: Move the point 2 units to the left and 1 unit down. The new coordinates will be (7 - 2, 8 - 1), which simplifies to (5, 7).
2. Rotation: To rotate the point 180° clockwise around the origin, we multiply the coordinates by the rotation matrix:
[x', y'] = [x cos(theta) - y sin(theta), x sin(theta) + y cos(theta)]
In this case, theta (θ) is 180°:
[x', y'] = [5 cos(180°) - 7 sin(180°), 5 sin(180°) + 7 cos(180°)]
Since cos(180°) = -1 and sin(180°) = 0, the coordinates simplify to:
[x', y'] = [-5 - 0, 5 + 7]
Therefore, the final position of point A will be A'(-5, 12).