Move point (3,50 degrees) according to the following symmetries (keep positive)
a. symmetrical to the origin
b. symmetrical to theta=0 degrees
c. symmetrical to theta= 90 degrees
To move a point according to symmetries, we need to understand what each symmetry means and how it affects the coordinates. Let's break down each case:
a. Symmetrical to the origin:
The symmetry to the origin is also called a reflection. It involves swapping the signs of both coordinates (x and y) with respect to the origin (0,0).
To apply this symmetry to the point (3, 50 degrees), we will take the negative of both coordinates to reflect it across the origin:
New x-coordinate = -3
New y-coordinate = -50 degrees
Therefore, the new point after reflecting across the origin would be (-3, -50 degrees).
b. Symmetrical to theta=0 degrees:
Symmetry with respect to a line passing through the origin and having an angle of 0 degrees (horizontal line) involves keeping the x-coordinate the same while taking the negative of the y-coordinate.
To apply this symmetry to the point (3, 50 degrees), we keep the x-coordinate as it is (3) and negate the y-coordinate:
New x-coordinate = 3
New y-coordinate = -50 degrees
Hence, the new point after reflecting with respect to theta=0 degrees would be (3, -50 degrees).
c. Symmetrical to theta=90 degrees:
Symmetry with respect to a line passing through the origin and having an angle of 90 degrees (vertical line) involves keeping the y-coordinate the same while taking the negative of the x-coordinate.
To apply this symmetry to the point (3, 50 degrees), we keep the y-coordinate as it is (50 degrees) and negate the x-coordinate:
New x-coordinate = -3
New y-coordinate remains the same = 50 degrees
Thus, the new point after reflecting with respect to theta=90 degrees would be (-3, 50 degrees).
Remember, when dealing with angles, it is common practice to keep the positive angle within the range of 0 to 360 degrees, so if the reflection yields a negative angle, we can add 360 degrees to it to get the equivalent positive angle within the range.