Final answer:
Rigid transformations such as rotation, reflection, or translation, can map one geometric figure onto another without altering distances or angles. The invariance of distance between points and to the origin under rotation is shown using the distance formula applied to the coordinates before and after rotation. Similarities between rotational and translational motion aid in solving related kinematics problems.
Step-by-step explanation:
The question seeks to determine which rigid transformation would map one triangle onto another. In this case, the rigid transformations include reflection, rotation, translation, or a combination of these. A rigid transformation is a transformation that preserves the distances and angles between points.
Show Invariance of Distance Under Rotation
Part (c) of the information provided relates to showing that the distance between two points remains the same under a rotation. To show this, we look at the initial coordinates of points P and Q and apply a rotation transformation. The equations for rotation in a coordinate system are:
x' = x cos q + y sin o
y' = -x sin p + y cos p
Using these formulas, we can find new coordinates for P and Q after rotation (P' and Q'), and calculate the distance using the distance formula. Since cosine and sine are functions that only affect the direction but not the magnitude, the calculated distance between P' and Q' will be the same as between P and Q, hence proving the invariance.
Invariance of Distance to the Origin Under Rotation
Part (b) refers to the distance from a point to the origin after a rotation. The relationship we're given is a variant of the Pythagorean theorem, suggesting that the distance from point P to the origin can be found by taking the square root of the sum of the squares of the coordinates, which is expressed as sqrt(x² + y²). After a rotation, this distance does not change.
Rotational and Translational Variables
In physics, there are similarities between rotational and translational motion. The quantities 0, w, and a for rotation correspond to x, v, and a for translation. Understanding these analogies helps in solving kinematics problems when either rotational or translational quantities are constant, as pointed out in Table 10.2.