Question

when a figure is transformed by a counterclockwise rotation about the origin, the x - and y - coordinates of its points change in predictable ways. \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the image of \\( \triangle a b c \\) after a 180 counterclockwise rotation about the origin. each point \\( ( x, y ) \\) is mapped to \\( ( - x, - y ) \\) : \\( a ( - 1,1 ) \
ightarrow a ^ { prime } ( 1, - 1 ) \\) \\( b ( - 2,3 ) \
ightarrow b ^ { prime } ( 2, - 3 ) \\) \\( c ( - 4,2 ) \
ightarrow c ^ { prime } ( 4, - 2 ) \\) describe the rotation that maps \\( \triangle q r s \\) to \\( \triangle q ^ { prime } r ^ { prime } s ^ { prime } \\). 1. write the coordinates of the vertices. \\( q ( \square, \square ) \
ightarrow q ^ { prime } ( \square, \square ) \\) \\( r ( \square, \square ) \
ightarrow r ^ { prime } ( \square, \square ) \\) \\( s ( \square, \square ) \
ightarrow s ^ { prime } ( \square, \square ) \\) 2. circle the transformation of \\( ( x, y ) \\) that follows the same pattern as the change in coordinates from \\( q \\) to \\( q ^ { prime } \\), \\( r \\) to \\( r ^ { prime } \\), and \\( s \\) to \\( s ^ { prime } \\). \\( ( x, y ) \
ightarrow ( - y, x ) \\) \\( ( x, y ) \
ightarrow ( - x, - y ) \\) \\( ( x, y ) \
ightarrow ( y, - x ) \\) 3. refer to the table at the top of the page. describe the rotation that maps \\( \triangle q r s \\) to \\( \triangle q ^ { prime } r ^ { prime } s ^ { prime } \\). on the back! 4. \\( \triangle f g h \\) has vertices \\( f ( 2,1 ) \\), \\( g ( 5,1 ) \\), and \\( h ( 5,4 ) \\). \\( \triangle f ^ { prime } g ^ { prime } h ^ { prime } \\) has vertices \\( f ^ { prime } ( - 1,2 ) \\), \\( g ^ { prime } ( - 1,5 ) \\), and \\( h ^ { prime } ( - 4,5 ) \\). describe the rotation that maps \\( \triangle f g h \\) to \\( \triangle f ^ { prime } g ^ { prime } h ^ { prime } \\).

Explanation:

Step1: Determine the coordinates

For point \(Q\): From the graph, \(Q(-1,1)\) and \(Q'( - 1,1)\) is incorrect. Wait, re - check. For the left - hand triangle (pre - image \(\triangle QRS\)):

• \(Q(-1,1)\), \(R(-1,3)\), \(S(-3,4)\)

For the right - hand triangle (image \(\triangle Q'R'S'\)):

• \(Q'(1,1)\), \(R'(3,1)\), \(S'(4,3)\)

Step2: Check the transformation rule

Let's take a general point \((x,y)\).
If we use the transformation \((x,y)\to(-y,x)\):

• For \(Q(-1,1)\): \((-1,1)\to(-1, - 1)\) (incorrect)

If we use \((x,y)\to(-x,-y)\):

• For \(Q(-1,1)\): \((-1,1)\to(1, - 1)\) (incorrect)

If we use \((x,y)\to(y, - x)\):

• For \(Q(-1,1)\): \((-1,1)\to(1,1)\)
• For \(R(-1,3)\): \((-1,3)\to(3,1)\)
• For \(S(-3,4)\): \((-3,4)\to(4,3)\)

Step3: Relate to the rotation table

From the table, when the transformation is \((x,y)\to(y, - x)\), the angle of rotation is \(270^{\circ}\) counter - clockwise about the origin.

Answer:

1. \(Q(-1,1)\to Q'(1,1)\), \(R(-1,3)\to R'(3,1)\), \(S(-3,4)\to S'(4,3)\)
2. The transformation \((x,y)\to(y, - x)\) is circled.
3. A \(270^{\circ}\) counter - clockwise rotation about the origin maps \(\triangle QRS\) to \(\triangle Q'R'S'\)
4. For \(\triangle FGH\) with \(F(2,1)\), \(G(5,1)\), \(H(5,4)\) and \(\triangle F'G'H'\) with \(F'(-1,2)\), \(G'(-1,5)\), \(H'(-4,5)\)

• Using the transformation rule \((x,y)\to(-y,x)\)
• For \(F(2,1)\): \((2,1)\to(-1,2)\)
• For \(G(5,1)\): \((5,1)\to(-1,5)\)
• For \(H(5,4)\): \((5,4)\to(-4,5)\)
• From the table, a \(90^{\circ}\) counter - clockwise rotation about the origin maps \(\triangle FGH\) to \(\triangle F'G'H'\)