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. 4. triangle hef is the image of triangle fgh after a 180 degree rotation around point k. select all statements that must be true. a. triangle hgf is congruent to triangle feh. b. triangle gfh is congruent to triangle efh. c. angle khe is congruent to angle khg. d. angle ghk is congruent to angle efk. e. segment eh is congruent to segment gh. f. segment hg is congruent to segment fe. g. segment fh is congruent to segment hf.

Explanation:

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Step1: Analyze translation

Translate \( XYZ \) using directed line segment \( YC \). This moves \( Y \) to \( C \).

Step2: Analyze rotation

Rotate \( X'Y'Z' \) (after translation) using \( C \) as the center. We want \( X' \) to coincide with \( B \).

Step3: Analyze reflection

Reflect \( X''Y''Z'' \) (after rotation) across line \( CB \). This will map the triangle \( XYZ \) (after translation and rotation) onto triangle \( BCA \).

• For congruence of triangles:
• A \( 180^{\circ}\) rotation is a rigid motion. Rigid motions preserve congruence. So, \(\triangle GFH\) (original) and \(\triangle EFH\) (image after \(180^{\circ}\) rotation) are congruent.
• For congruence of angles:
• Since it is a \(180^{\circ}\) rotation about \(K\), \(\angle KHE\) and \(\angle KHG\) are vertical - like angles formed by the rotation (if we consider the rotation mapping \(H\) to \(H\) (in a sense of the rotation property around \(K\)) and the other points). Also, \(\angle GHK\) and \(\angle EF K\) are congruent as corresponding angles of congruent triangles \(\triangle GFH\) and \(\triangle EFH\).
• For congruence of segments:
• Segments \(EH\) and \(GH\) are corresponding sides of congruent triangles \(\triangle GFH\) and \(\triangle EFH\). Segments \(HG\) and \(FE\) are corresponding sides of congruent triangles \(\triangle GFH\) and \(\triangle EFH\).

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 \( CB \).

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