Question

name:
pd.
date:
geometry chapter 4 review

1. graph (\triangle abc) with vertices (a(2,1)), (b(-1,2)) and (c(-2,-1)) and its image after the following sequence of transformations.

translation: ((x,y)\to(x - 1,y - 3))
translation: ((x,y)\to(x + 4,y + 3))

Explanation:

Step1: Apply the first translation

For point \(A(2,1)\):
\(x = 2-1=1\), \(y = 1 - 3=-2\), new point \(A_1(1,-2)\)
For point \(B(-1,2)\):
\(x=-1 - 1=-2\), \(y=2-3=-1\), new point \(B_1(-2,-1)\)
For point \(C(-2,-1)\):
\(x=-2-1=-3\), \(y=-1 - 3=-4\), new point \(C_1(-3,-4)\)

Step2: Apply the second translation to the points from Step1

For point \(A_1(1,-2)\):
\(x = 1+4=5\), \(y=-2 + 3=1\), final point \(A_2(5,1)\)
For point \(B_1(-2,-1)\):
\(x=-2 + 4=2\), \(y=-1+3=2\), final point \(B_2(2,2)\)
For point \(C_1(-3,-4)\):
\(x=-3+4=1\), \(y=-4 + 3=-1\), final point \(C_2(1,-1)\)

Answer:

The original triangle \(\triangle ABC\) has vertices \(A(2,1)\), \(B(-1,2)\), \(C(-2,-1)\). After the first translation \((x,y)\to(x - 1,y - 3)\), the vertices are \(A_1(1,-2)\), \(B_1(-2,-1)\), \(C_1(-3,-4)\). After the second translation \((x,y)\to(x + 4,y + 3)\), the vertices of the final image \(\triangle A_2B_2C_2\) are \(A_2(5,1)\), \(B_2(2,2)\), \(C_2(1,-1)\). To graph:

• Plot \(A(2,1)\), \(B(-1,2)\), \(C(-2,-1)\) for \(\triangle ABC\).
• Then plot \(A_2(5,1)\), \(B_2(2,2)\), \(C_2(1,-1)\) for the translated triangle. Connect the respective sets of three points with line - segments.