Question

1. the triangles are congruent. which sequence of rigid motions will take triangle xyz onto triangle bca?

a. translate xyz using directed line segment yc. rotate xyz using c as the center so that x coincides with b. reflect xyz across line cb.
b. translate xyz using directed line segment yc. rotate xyz using c as the center so that x coincides with b. reflect xyz across line ac.
c. translate xyz using directed line segment yc. rotate xyz using c as the center so that x coincides with a. reflect xyz across line cb.
d. translate xyz using directed line segment yc. rotate xyz using c as the center so that x coincides with a. reflect xyz across line ac.

Explanation:

Step1: Analyze translation

First, we need to move triangle $XYZ$ so that one of its points gets close to the corresponding point on triangle $BCA$. Translating $XYZ$ using directed - line segment $YC$ will move point $Y$ to point $C$.

Step2: Analyze rotation

After translation, we rotate the translated triangle $X'Y'Z'$ about point $C$. We want to align one of the sides. To map triangle $XYZ$ onto $BCA$, we rotate so that $X'$ coincides with $B$.

Step3: Analyze reflection

After rotation, we reflect the rotated triangle $X''Y''Z''$ across line $CB$ to map it exactly onto triangle $BCA$.

Answer:

A. Translate $XYZ$ using directed line segment $YC$. Rotate $X'Y'Z'$ using $C$ as the center so that $X'$ coincides with $B$. Reflect $X''Y''Z''$ across line $CB$.