Question

point reflections n-gen math geometry one of the most important transformations is a 180 rotation about a point. these types of rotations are known as point reflections. 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{ab} \\) 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{ab} \\) and \\( \overline{ab} \\)? why? exercise #2: using a straightedge and compass only, construct the image of \\( \triangle efg \\) after a point reflection across point m. label the image \\( \triangle efg \\). 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 mno \\) after a reflection across the origin. label its image \\( \triangle mno \\). show the mapping below. (b) give a general rule for reflecting a point across the origin. \\( f:(x,y) \to \\)

Explanation:

Step1: Find the mapping of each vertex

For a point \((x,y)\) reflected across the origin \((0,0)\) (a \(180^{\circ}\) rotation about the origin), the rule is \((x,y)\to(-x,-y)\).
For vertex \(M(1,4)\):
\(M(1,4)\to M'(- 1,-4)\)
For vertex \(N(4,6)\):
\(N(4,6)\to N'(-4,-6)\)
For vertex \(O(8,2)\):
\(O(8,2)\to O'(-8,-2)\)

Step2: General rule

The general rule for reflecting a point \((x,y)\) across the origin is a transformation that changes the sign of both the \(x\) - coordinate and the \(y\) - coordinate.

Answer:

(a) The mapping is \(M(1,4)\to M'(-1,-4)\), \(N(4,6)\to N'(-4,-6)\), \(O(8,2)\to O'(-8,-2)\)
(b) \(f:(x,y)\to(-x,-y)\)