If
�
�
‾
=
AB
=
11
11 inches then find the distance of
�
′
�
′
‾
A
′
B
′
after a reflection over the x-axis and a rotation of
162
°
162° clockwise.
To find the distance of A' B' after a reflection over the x-axis and a rotation of 162° clockwise, we can follow these steps:
Step 1: Reflection over the x-axis
This changes the sign of the y-coordinate, while keeping the x-coordinate the same. Since A is at (0, 11), its reflection A' will be at (0, -11).
Since B is at (11, 0), its reflection B' will be at (11, 0).
Step 2: Rotation of 162° clockwise
To rotate a point (x, y) clockwise by θ, the new coordinates (x', y') after rotation can be found using the following formulas:
x' = x*cos(θ) + y*sin(θ)
y' = -x*sin(θ) + y*cos(θ)
In this case, we want to rotate the points A'(-11, 0) and B'(0, -11) by 162° clockwise.
Let's calculate the new coordinates:
For point A'(-11, 0):
x' = -11*cos(162°) + 0*sin(162°) ≈ -0.8535
y' = -(-11)*sin(162°) + 0*cos(162°) ≈ 9.5624
So the new coordinates for A' after rotation are approximately (-0.8535, 9.5624).
For point B'(0, -11):
x' = 0*cos(162°) + (-11)*sin(162°) ≈ -9.5624
y' = -0*sin(162°) + (-11)*cos(162°) ≈ -0.8535
So the new coordinates for B' after rotation are approximately (-9.5624, -0.8535).
Step 3: Calculate the distance between A' and B'
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distance between the new points A'(-0.8535, 9.5624) and B'(-9.5624, -0.8535):
distance = √((-9.5624 - (-0.8535))^2 + (-0.8535 - 9.5624)^2)
= √((-8.7089)^2 + (-10.4159)^2)
≈ √(75.8287 + 108.4526)
≈ √184.2813
≈ 13.5790
Therefore, the distance of A' B' after the reflection over the x-axis and rotation of 162° clockwise is approximately 13.5790.