Question

graph \\( \triangle a b c \\) with vertices \\( a ( 4,2 ) \\), \\( b ( 0, - 2 ) \\), and \\( c ( - 2,0 ) \\) its image after the similarity transformation
rotation: \\( 90 ^ { \circ } \\) clockwise about the origin.
dilation: \\( ( x , y ) \
ightarrow ( 2 x , 2 y ) \\)

Explanation:

Step1: Find the rotation coordinates

The rule for a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, -x)\).
For \(A(4,2)\): After rotation, \(A'(2,-4)\).
For \(B(0,-2)\): After rotation, \(B'(-2,0)\).
For \(C(-2,0)\): After rotation, \(C'(0,2)\).

Step2: Find the dilation coordinates

The dilation rule is \((x,y)\to(2x,2y)\).
For \(A'(2,-4)\): After dilation, \(A''(2\times2,2\times(-4))=(4,-8)\).
For \(B'(-2,0)\): After dilation, \(B''(2\times(-2),2\times0)=(-4,0)\).
For \(C'(0,2)\): After dilation, \(C''(2\times0,2\times2)=(0,4)\).

Answer:

The vertices of the image \(\triangle A''B''C''\) are \(A''(4, - 8)\), \(B''(-4,0)\) and \(C''(0,4)\)