Question

rotations
for items 1 - 2, use \\( \triangle b c d \\) in the figure shown.

1. what are the vertices of the image \\( r _ { ( 90 ^ { \circ }, 0 ) } ( \triangle b c d ) \\) ?

(a) \\( b ^ { \prime } ( - 3 , - 3 ) , c ^ { \prime } ( - 1,4 ) , d ^ { \prime } ( 2,0 ) \\)
(b) \\( b ^ { \prime } ( 3,3 ) , c ^ { \prime } ( 1 , - 4 ) , d ^ { \prime } ( - 2,0 ) \\)
(c) \\( b ^ { \prime } ( 3 , - 3 ) , c ^ { \prime } ( 1,4 ) , d ^ { \prime } ( 0,2 ) \\)
(d) \\( b ^ { \prime } ( - 3 , - 3 ) , c ^ { \prime } ( - 1 , - 4 ) , d ^ { \prime } ( 0 , - 2 ) \\)

1. select all the sequences of reflections that produce an image equivalent to the

image \\( r _ { ( 180 ^ { \circ }, 0 ) } ( \triangle b c d ) \\).
\\( \square \\) a. \\( ( r _ { x - \text { axis } } cdot r _ { y - \text { axis } } ) ( \triangle b c d ) \\)
\\( \square \\) b. \\( ( r _ { y - \text { axis } } cdot r _ { x - \text { axis } } ) ( \triangle b c d ) \\)
\\( \square \\) c. \\( ( r _ { x - \text { axis } } cdot r _ { x - \text { axis } } ) ( \triangle b c d ) \\)
\\( \square \\) d. \\( ( r _ { y - \text { axis } } cdot r _ { y - \text { axis } } ) ( \triangle b c d ) \\)
\\( \square \\) e. \\( ( r _ { y = x } cdot r _ { x - \text { axis } } ) ( \triangle b c d ) \\)

1. \\( overline { a b } \\) is rotated \\( 120 ^ { \circ } \\) clockwise about \\( b \\). then \\( overline { a b } \\) is rotated \\( 45 ^ { \circ } \\) counterclockwise

about \\( a \\). what is the image of \\( a \\) as a composition of transformations?
(a) \\( ( r _ { ( 120 ^ { \circ }, b ) } cdot r _ { ( - 45 ^ { \circ }, a ) } ) ( a ) \\)
(b) \\( ( r _ { ( - 45 ^ { \circ }, a ) } cdot r _ { ( 120 ^ { \circ }, b ) } ) ( a ) \\)
(c) \\( ( r _ { ( - 120 ^ { \circ }, b ) } cdot r _ { ( 45 ^ { \circ }, a ) } ) ( a ) \\)
(d) \\( ( r _ { ( 45 ^ { \circ }, a ) } cdot r _ { ( - 120 ^ { \circ }, b ) } ) ( a ) \\)

1. suppose \\( r _ { ( 140 ^ { \circ }, p ) } ( a ) = b \\) and \\( ( r _ { overrightarrow { p d } } cdot r _ { overrightarrow { p c } } ) ( a ) = b \\). what is \\( m \angle c p d \\) ?

1. how many times does the rotation \\( r _ { ( 120 ^ { \circ }, p ) } \\) need to be applied to a figure to

map the figure onto itself?

Answer:

1. Step - by - Step Format for Rotation of Points:

• First, find the coordinates of \(B\), \(C\), and \(D\) from the graph. Let's assume \(B(-3,3)\), \(C(4,1)\), \(D(0, - 2)\).
• The rule for a rotation of \(90^{\circ}\) clockwise about the origin \((x,y)\to(y, - x)\).
• For point \(B(-3,3)\):
• Using the formula \((x,y)\to(y, - x)\), when \(x=-3\) and \(y = 3\), we get \(B'(3,3)\).
• For point \(C(4,1)\):
• Using the formula \((x,y)\to(y, - x)\), when \(x = 4\) and \(y=1\), we get \(C'(1,-4)\).
• For point \(D(0,-2)\):
• Using the formula \((x,y)\to(y, - x)\), when \(x = 0\) and \(y=-2\), we get \(D'(-2,0)\).

1. Answer - Explanation Format for Reflection - Rotation Equivalence:

• Brief Explanations:
• The rule for a rotation of \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\).
• For a reflection over the \(x\) - axis \((x,y)\to(x, - y)\) and then over the \(y\) - axis \((x, - y)\to(-x,-y)\). So, \((r_{x - axis}\circ r_{y - axis})(\triangle BCD)\) (composition of reflections \(r_{x - axis}\) followed by \(r_{y - axis}\)) gives the same result as a \(180^{\circ}\) rotation about the origin.
• Similarly, for a reflection over the \(y\) - axis \((x,y)\to(-x,y)\) and then over the \(x\) - axis \((-x,y)\to(-x,-y)\). So, \((r_{y - axis}\circ r_{x - axis})(\triangle BCD)\) gives the same result as a \(180^{\circ}\) rotation about the origin.
• For \((r_{x - axis}\circ r_{x - axis})(\triangle BCD)\), \((x,y)\to(x, - y)\to(x,y)\) (it is the identity transformation).
• For \((r_{y - axis}\circ r_{y - axis})(\triangle BCD)\), \((x,y)\to(-x,y)\to(x,y)\) (it is the identity transformation).
• For \((r_{y=x}\circ r_{x - axis})(\triangle BCD)\), the rule for reflection over \(y = x\) is \((x,y)\to(y,x)\) and then reflection over \(x\) - axis \((y,x)\to(y, - x)\) which is not equivalent to a \(180^{\circ}\) rotation.
• Answer:
• A. \((r_{x - axis}\circ r_{y - axis})(\triangle BCD)\)
• B. \((r_{y - axis}\circ r_{x - axis})(\triangle BCD)\)

1. Answer - Explanation Format for Composition of Rotations:

• Brief Explanations:
• A clockwise rotation of \(120^{\circ}\) about \(B\) is equivalent to \(R_{(- 120^{\circ},B)}\) (since clockwise rotations are negative in the standard rotation formula \(R_{(\theta,O)}\) where \(\theta\) is measured counter - clockwise). A counter - clockwise rotation of \(45^{\circ}\) about \(A\) is \(R_{(45^{\circ},A)}\).
• When we compose transformations, the first transformation is the one that is applied first. The segment \(\overline{AB}\) is first rotated \(120^{\circ}\) clockwise about \(B\) (i.e., \(R_{(-120^{\circ},B)}\)) and then \(45^{\circ}\) counter - clockwise about \(A\) (i.e., \(R_{(45^{\circ},A)}\)). In function composition \((f\circ g)(x)=f(g(x))\), so the composition is \((R_{(-120^{\circ},B)}\circ R_{(45^{\circ},A)})(A)\).
• Answer:
• C. \((R_{(-120^{\circ},B)}\circ R_{(45^{\circ},A)})(A)\)

1. Step - by - Step Format for Angle of Rotation and Reflection:

• Brief Explanations:
• By the theorem of rotation as a composition of reflections: If \(R_{(\theta,P)}(A)=B=(r_{\overrightarrow{PD}}\circ r_{\overrightarrow{PC}})(A)\), then the measure of the angle of rotation \(\theta = 2m\angle CPD\).
• Given \(R_{(140^{\circ},P)}(A)=B=(r_{\overrightarrow{PD}}\circ r_{\overrightarrow{PC}})(A)\).
• Using the formula \(\theta = 2m\angle CPD\), we can solve for \(m\angle CPD\).
• Step1: Use the rotation - reflection angle formula
• We know that for a rotation \(R_{(\theta,P)}\) which is a composition of two reflections \(r_{\overrightarrow{PD}}\) and \(r_{\overrightarrow{PC}}\) about lines \(\overrightarrow{PC}\) and \(\overrightarrow{PD}\) respectively, \(\theta=2m\angle CPD\).
• Step2: Solve for \(m\angle CPD\)
• Given \(\theta = 140^{\circ}\), then \(m\angle CPD=\frac{\theta}{2}\).
• Substitute \(\theta = 140^{\circ}\) into the formula: \(m\angle CPD=\frac{140^{\circ}}{2}=70^{\circ}\).
• Answer: \(70^{\circ}\)

1. Step - by - Step Format for Rotation Mapping onto Itself:

• Brief Explanations:
• We want to find the smallest positive integer \(n\) such that \(n\times120^{\circ}=360^{\circ}k\) (where \(k\) is a positive integer) because a full rotation (a multiple of \(360^{\circ}\)) maps a figure onto itself.
• Step1: Set up the equation
• Let \(n\times120^{\circ}=360^{\circ}\) (the smallest non - zero multiple of \(360^{\circ}\) we can start with).
• Step2: Solve for \(n\)
• \(n=\frac{360^{\circ}}{120^{\circ}}\).
• \(n = 3\) (since \(3\times120^{\circ}=360^{\circ}\)).
• Answer: \(3\)

1. Answer: B. \(B'(3,3),C'(1, - 4),D'(-2,0)\)