When a polygon undergoes a translation on a coordinate plane, several properties of the polygon are preserved. Here is a step-by-step detailed solution to determine which statement is not necessarily true when polygon GHIJ translates 8 units to the left to form polygon G'H'I'J':
1. Understanding Translation:
- Translation is a type of transformation that slides each point of a shape a certain distance in a given direction. In this case, each point of polygon GHIJ is translated 8 units to the left.
2. Analyzing Each Statement:
- Statement A: [tex]\( GH = G' H' \)[/tex]
- Translation preserves the lengths of the sides of the polygon. This means that the distance between points G and H will be the same as the distance between the translated points G' and H'.
- Therefore, [tex]\( GH = G' H' \)[/tex] is true.
- Statement B: [tex]\( G'G = 8 \)[/tex] units
- The translation distance is given as 8 units to the left. This means that the distance between the original position G and the translated position G' will be exactly 8 units.
- Therefore, [tex]\( G'G = 8 \)[/tex] units is true.
- Statement C: [tex]\( m \angle HIJ = m \angle H'I'J' \)[/tex]
- Translation preserves the angles of a polygon. The measure of angle [tex]\( HIJ \)[/tex] will be the same as the measure of the corresponding angle [tex]\( H'I'J' \)[/tex].
- Therefore, [tex]\( m \angle HIJ = m \angle H'I'J' \)[/tex] is true.
- Statement D: [tex]\( m \angle HI'J = m \angle H'IJ \)[/tex]
- This statement suggests that the angle between points [tex]\( H, I' \)[/tex], and [tex]\( J \)[/tex] is equal to the angle between [tex]\( H', I \)[/tex], and [tex]\( J \)[/tex]. Since [tex]\( I' \)[/tex] and [tex]\( I \)[/tex] represent different corresponding points after translation, there is no guarantee that [tex]\( m \angle HI'J = m \angle H'IJ \)[/tex].
- This correlation does not necessarily hold true under translation.
3. Conclusion:
- Statements A, B, and C are true given the properties of translation.
- Statement D is the one that is not necessarily true.
Therefore, the equation that is not necessarily true is [tex]\( D \)[/tex].
