Question

point reflections
n-gen math* geometry
point reflections
one of the most important transformations is a 180 rotation about a point. these types of rotations are known
as point reflections.
a point reflection is a rotation of points in the plane about a given point by 180°. since it is exactly one-
half of a full rotation, the direction of rotation does not matter.
exercise #1: using a straightedge and a compass only, find the image of \\( \overline { a b } \\) after a point reflection across or
through point c. again, this is the same as a rotation about point c by 180°.
what must be true about the lengths \\( \overline { a b } \\)
and \\( \overline { a ^ { \prime } b ^ { \prime } } \\) ? why?
exercise #2: using a straightedge and compass only, construct
the image of \\( \triangle e f g \\) after a point reflection across point m.
leave all construction marks. label the image \\( \triangle e ^ { \prime } f ^ { \prime } g ^ { \prime } \\).
we can certainly have point reflections in the coordinate plane. the most common is a reflection across the origin
(i.e. a 180° rotation about the origin).
exercise #3: triangle mno is graphed with vertices at m(1,4), n(4,6), and
o(8,2).
(a) draw the image of \\( \triangle m n o \\) after a reflection across the origin. label its
image \\( \triangle m ^ { \prime } n ^ { \prime } o ^ { \prime } \\). show the mapping below.
(b) give a general rule for reflecting a point across the origin.
\\( f : ( x , y ) \
ightarrow \\)

Explanation:

Step1: Understand the property of point reflection

A point reflection (or \(180^{\circ}\) rotation) is a rigid transformation. Rigid transformations preserve the lengths of line - segments.

Step2: Apply the property to \(\overline{AB}\) and \(\overline{A'B'}\)

Since a point reflection (a \(180^{\circ}\) rotation about a point \(C\)) is a rigid transformation, for any line - segment \(\overline{AB}\) and its image \(\overline{A'B'}\) after a point reflection, the lengths are equal. Mathematically, if we consider the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let \(A=(x_1,y_1)\), \(B=(x_2,y_2)\) and after a \(180^{\circ}\) rotation about a point \(C=(a,b)\), \(A'=(2a - x_1,2b - y_1)\) and \(B'=(2a - x_2,2b - y_2)\).

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Answer:

The lengths of \(\overline{AB}\) and \(\overline{A'B'}\) are equal. Because a point reflection (a \(180^{\circ}\) rotation about a point) is a rigid transformation, and rigid transformations preserve the lengths of line - segments.