Question

1. draw and label the image of \\( \overline { c d } \\) under the following sequence of rigid motions.

• \\( 90 ^ { \circ } \\) clockwise rotation around point \\( c \\)
• translation along \\( \overrightarrow { a b } \\)

Explanation:

Step1: Rotate segment \( \overline{CD} \)

A \( 90^{\circ} \) clockwise rotation around point \( C \). Use the rule for a \( 90^{\circ} \) clockwise rotation: \((x,y)\to(y, -x)\) (assuming \( C \) is the origin for the local coordinate system of the segment). Let \( D=(x,y) \) relative to \( C \). After rotation, \( D'=(y, -x) \) relative to \( C \). Draw \( \overline{CD'} \).

Step2: Translate the rotated segment

Translation along \( \overrightarrow{AB} \). A translation moves every point of a figure or space by the same distance in a given direction. Measure the vector \( \overrightarrow{AB} \) (the distance and direction from \( A \) to \( B \)). Apply this vector to both \( C \) and \( D' \). If \( C=(c_x,c_y) \) and \( \overrightarrow{AB}=(b_x - a_x,b_y - a_y) \), then the new \( C''=(c_x+(b_x - a_x),c_y+(b_y - a_y)) \) and \( D''=(y + c_x+(b_x - a_x),-x + c_y+(b_y - a_y)) \). Draw \( \overline{C''D''} \).

Answer:

Follow the two - step geometric transformation (rotation then translation) as described above to draw the final image of \( \overline{CD} \).