Question

geometry - composition of transformations
graph each figure with the given vertices and its ima

1. (\triangle abc:a(1, - 3),b(2,0),c(3, - 1);t_{-2,1}circ r_{o,180^{circ}})

Explanation:

Step1: Find the coordinates after rotation

The rule for a \(180^{\circ}\) rotation about the origin \(R_{O,180^{\circ}}\) is \((x,y)\to(-x,-y)\).
For point \(A(1, - 3)\): \((1,-3)\to(-1,3)\)
For point \(B(2,0)\): \((2,0)\to(-2,0)\)
For point \(C(3,-1)\): \((3,-1)\to(-3,1)\)

Step2: Find the coordinates after translation

The rule for the translation \(T_{-2,1}\) is \((x,y)\to(x - 2,y + 1)\)
For the rotated point \(A'(-1,3)\): \((-1-2,3 + 1)=(-3,4)\)
For the rotated point \(B'(-2,0)\): \((-2-2,0 + 1)=(-4,1)\)
For the rotated point \(C'(-3,1)\): \((-3-2,1 + 1)=(-5,2)\)

Answer:

The vertices of \(\triangle ABC\) are \(A(1,-3)\), \(B(2,0)\), \(C(3,-1)\). The vertices of the image \(\triangle A''B''C''\) after \(T_{-2,1}\circ R_{O,180^{\circ}}\) are \(A''(-3,4)\), \(B''(-4,1)\), \(C''(-5,2)\). To graph:

• Plot \(A(1,-3)\), \(B(2,0)\), \(C(3,-1)\) for \(\triangle ABC\).
• Plot \(A''(-3,4)\), \(B''(-4,1)\), \(C''(-5,2)\) for the image triangle. Connect the respective sets of three points to form the two triangles.