Final answer:
The invariant point under the combined transformation of a rotation and translation is (1,0), corresponding to option (d). The invariant point does not change position after the transformation, which in this case consists of a 180° rotation about the point (-1,0), followed by a translation of (-3/2, 0).
Step-by-step explanation:
The question involves finding the invariant point of a combined transformation consisting of a 180° rotation about the point (-1,0) followed by a translation with a vector (-3/2, 0). An invariant point under a transformation is a point that doesn't change its position after the transformation is applied.
To find the invariant point, we need to understand what happens to a point when these transformations are applied. A 180° rotation about the point (-1,0) would flip the point across that center, effectively changing its x coordinate's sign while keeping distance from (-1,0) constant. A subsequent translation by (-3/2, 0) would shift every point on the plane left by 3/2 units
Let's start by considering option (a) (-1,0). After a 180° rotation about this very point, its position remains unchanged (since it is the center of rotation). However, after the translation left by 3/2 units, its x-value would become -1 - 3/2 = -2.5, which would no longer be invariant. Therefore, (-1,0) is not the invariant point.
Looking at option (b) (-3/2,0), a 180° rotation around (-1,0) would invert the x-coordinate in reference to the center of rotation, giving us (-1 - (-3/2 + 1)) = (-1 + 1/2) = (-1/2, 0). A translation by (-3/2, 0) would then move the point (-1/2, 0) left by 3/2 units, resulting in a new point at (-1/2 - 3/2, 0) = (-2, 0), which also isn't the same as the starting point.
However, options (c) and (d) are a little different. Performing the mentioned rotation on point (c) (0,0), we would find it lands at a new point relative to the center (-1,0) but translations move every point left by the same amount, so an invariant point after translation should be to the right before translation. Thus, (0,0) is also not correct. Regarding option (d) (1,0), its 180° rotation about (-1,0) would result in (-3,0), and then a translation of (-3/2, 0) returns it to its original position at (1,0).
Thus, the coordinates of the invariant point after a 180° rotation about the point (-1,0) followed by a translation with a vector (-3/2, 0) are (1,0), corresponding to option (d).