Answer:
ia) center: (0, 0), the origin
ib) angle: 90°
ic) direction: CCW
ii) LM ≅ L'M', ∠L ≅ ∠L'
iii) (2, 1)
Explanation:
A rigid transformation preserves size and shape of a figure, so the image is congruent to the original. The rigid transformations are rotation, reflection, and translation. Each has characteristics that allow one to determine the nature of the transformation(s) involved.
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rotation
angle and direction
One way to determine the angle and direction of rotation of a figure is to compare a horizontal segment on the original with the corresponding segment on the image. Here, a convenient segment is NM, which is horizontal and points to the right. Its angle relative to the direction of the +x axis is 0°.
The transformed segment N'M' points upward, at an angle relative to the +x axis of 90°. This tells you the figure was rotated 90° in the CCW direction. (This is fully equivalent to a rotation of 270° in the CW direction.)
center
Points in a rotated image always remain at the same distance from the center of rotation as the original points. In most cases, the center of rotation is the origin. We can check to see if that is the case by looking at the distances of a couple of points from the origin.
N is at coordinates (1, 1), so is √(1² +1²) = √2 from the origin
N' is at coordinates (-1, 1), so is √((-1)² +1²) = √2 from the origin
L is at coordinates (1, 3), so is √(1² +3²) = √10 from the origin
L' is at coordinates (-3, 1) so is √((-3)² +1²) = √10 from the origin
This confirms that the origin is the center of rotation.
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Since each point and its image are on a circle centered at the center of rotation, the line between them is a chord of that circle. The perpendicular bisector of that chord goes through the center of rotation. Here, we observe that the perpendicular bisector of NN' is the y-axis. The perpendicular bisector of LL' is a line with slope -2 through (-1, 2). It will intersect the y-axis at the origin, confirming the origin as the center of rotation.
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geometric relationships
A rigid transformation preserves lengths and angles, so any length or angle on the rotated figure is congruent to the original:
LM ≅ L'M', MN ≅ M'N', NL ≅ N'L'
∠L ≡ ∠L', ∠M ≡ ∠M', ∠N ≡ ∠N'
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translation
The components of a translation vector add to the corresponding coordinates of a point.
L +v = L'
(1, 3) +(1, -2) = (2, 1) . . . image of point L