Question

a sequence of transformations is described below.

• a translation
• a reflection over a horizontal line \overleftrightarrow{pq}
• a vertical stretch about \overleftrightarrow{pq}

which of the following must be preserved under this sequence of transformations?
choose 1 answer:
a angle measures only
b segment lengths only
c both angle measures and segment lengths
d neither angle measures nor segment lengths

Explanation:

Step1: Analyze Translation

Translation is a rigid transformation. In a translation, both angle measures and segment lengths are preserved. So after translation, angle measures and segment lengths remain the same.

Step2: Analyze Reflection

Reflection over a horizontal line \( \overleftrightarrow{PQ} \) is also a rigid transformation. Rigid transformations (like reflection) preserve both angle measures and segment lengths. So after reflection, angle measures and segment lengths are still preserved.

Step3: Analyze Vertical Stretch

A vertical stretch about \( \overleftrightarrow{PQ} \) is a non - rigid transformation. A vertical stretch changes the length of segments that are not parallel to the line of stretch (in this case, vertical segments will be stretched). However, angle measures are preserved under a vertical stretch. Because when we stretch a figure vertically, the angles between the sides of the figure do not change. For example, if we have a right - angled triangle and we perform a vertical stretch, the right angle will still be a right angle, and other angles will also remain the same. But the lengths of the vertical sides (and sides related to them) will change. So after the vertical stretch, angle measures are preserved but segment lengths are not.
Combining the three transformations: translation (preserves both), reflection (preserves both), and vertical stretch (preserves angles, changes lengths). So overall, angle measures are preserved throughout the sequence of transformations, but segment lengths are not preserved in the vertical stretch step.

Answer:

A. Angle measures only