Question

The vertices of a rectangle are M(-5, -5), N(-1, -5), O(-1, 1), and P(-5, 1). A rotation about the origin maps P to the point (1, 5). How many degrees was the rotation?

a) 45 degrees
b) 90 degrees
c) 180 degrees
d) 270 degrees

Answer 1

The rectangle was rotated by 90 degrees about the origin to map vertex P from (-5, 1) to (1, 5). The distance from point P to the origin remains invariant under rotations of the coordinate system, as illustrated by the equal distances before and after rotation.

Step-by-step explanation:

The student asked about the number of degrees a rectangle must be rotated about the origin to map a vertex P from (-5, 1) to (1, 5). Given the coordinates of P and its image after rotation, we can determine the angle of rotation. The original point P(-5, 1) maps to the point (1, 5) after rotation, which suggests a counterclockwise quarter turn about the origin. Hence, since a quarter turn is equivalent to 90 degrees, that is the angle of the rotation.

Furthermore, to show that the distance from point P to the origin is invariant under rotation, we calculate the distance before and after the rotation. The distance from P to the origin before rotation is √((-5)² + (1)²) = √(26), and the distance from the image of P to the origin after rotation is √((1)² + (5)²) = √(26), confirming the invariance of distance.