Question

4 a rotate 270° cw from the origin call it hat h 2,2 h 2,2 a 7,-6 a 7,6 t 5,-9 t 5,9 b reflect over the line x = 1. call it hat h __ a t c translate 3 right and 4 down. call it bug b u g __ is congruency preserved? is orientation preserved? translation reflection rotation

Explanation:

Step1: Determine coordinates after rotation

The rule for a \(270^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\).

• For \(H(1, - 1)\), \(H'( - 1,-1)\)
• For \(A(4, - 3)\), \(A'( - 3,-4)\)
• For \(T(3, - 6)\), \(T'( - 6,-3)\)

Step2: Determine coordinates after reflection over \(x = 1\)

The rule for reflection over the line \(x=a\) is \((x,y)\to(2a - x,y)\). Here \(a = 1\), so \((x,y)\to(2 - x,y)\)

• For \(H'( - 1,-1)\), \(H''(2-(-1),-1)=(3,-1)\)
• For \(A'( - 3,-4)\), \(A''(2-(-3),-4)=(5,-4)\)
• For \(T'( - 6,-3)\), \(T''(2-(-6),-3)=(8,-3)\)

Step3: Determine coordinates after translation

The rule for translation \(3\) units right and \(4\) units down is \((x,y)\to(x + 3,y-4)\)

• For \(H''(3,-1)\), \(B(3 + 3,-1-4)=(6,-5)\)
• For \(A''(5,-4)\), \(U(5 + 3,-4-4)=(8,-8)\)
• For \(T''(8,-3)\), \(G(8+3,-3 - 4)=(11,-7)\)

Step4: Check congruency and orientation

• Congruency: Translation, reflection, and rotation are all rigid transformations. Rigid transformations preserve the shape and size of the figure. So for translation: yes, reflection: yes, rotation: yes.
• Orientation: Translation and rotation preserve orientation (the order of vertices in a figure remains the same in a clock - wise or counter - clockwise sense). Reflection changes orientation (a clock - wise figure becomes counter - clockwise and vice - versa). So for translation: yes, reflection: no, rotation: yes.

Answer:

• \(H''=(3,-1)\), \(A''=(5,-4)\), \(T''=(8,-3)\)
• \(B=(6,-5)\), \(U=(8,-8)\), \(G=(11,-7)\)
• Congruency: Translation - yes, Reflection - yes, Rotation - yes
• Orientation: Translation - yes, Reflection - no, Rotation - yes