Exercises

1. Describe each of the following translations.
   1. 
   2. 
   3. 
   4. Quadrilateral \(ABCD\) with vertices \(A(-10,-6)\), \(B(-10,3)\), \(C(-4,4)\), and \(D(-4,-9)\) is translated to \(A'(0,-2)\), \(B'(0,7)\), \(C'(6,8)\), and \(D'(6,-5)\).
2. Graph the image of each polygon under the translation described.  State the coordinates of the vertices of the image.
   1. Translate \(\triangle ABC\) \(4\) units to the right and \(2\) units down.
      
   2. Translate quadrilateral \(JKLM\) \(4\) units to the left and \(6\) units down. 
   3. Translate pentagon \(PQRST\), with vertices \(P(-1,4)\), \(Q(5,3)\), \(R(6,0)\), \(S(4,-2)\), and \(T(1,-1)\),
      \((-9)\) units in the \(x\)-direction and \(2\) units in the \(y\)-direction. 

3. \(\triangle TAG\) is translated \(8\) units in the \(x\)-direction and \((-1)\) units in the \(y\)-direction.
   
   The vertices of the image, \(\triangle T'A'G'\) are \(T'(4,5)\), \(A'(9,5)\), and \(G'(4,0)\). 
   
   What are the coordinates of vertices \(T\), \(A\), and \(G\) ?

   
4. \(\triangle A'B'C'\) is the image of \(\triangle ABC\) under a translation. The vertices of the image, \(\triangle A'B'C'\) are \(A'(3,3)\), \(B'(5,1)\), and \(C'(1,0)\).
   1. If \(A'\) is the image of \(A(-3,0)\), describe the translation.
   2. Graph the original figure.
5. \(\triangle RST\) is translated \(3\) units in the \(x\)-direction and \((-5)\) units in the \(y\)-direction to produce \(\triangle R'S'T'\).
   Then \(\triangle R'S'T'\) is translated \((-4)\) units in the \(x\)-direction and \(9\) units in the \(y\)-direction to produce \(\triangle R''S''T''\). The coordinates of one vertex in each triangle are given here:  \(R(1,2)\), \(S'(-1,-5)\), and \(T''(-3,1)\).
   Determine the coordinates of all the other vertices of the three triangles.
6. Mark each statement as true or false. If the statement is false, then explain why.
   1. A polygon and its image under a translation are congruent.
   2. A translation that transforms \(A(2,5)\) to \(A'(-3,4)\) will transform \(B(-4,3)\) to \(B'(-8,2)\).
   3. A translation is a transformation in which every point of an object moves in the same direction by the same distance. 
   4. If the area of \(\triangle ABC\) is equal to the area of \(\triangle DEF\), then \(\triangle DEF\) is a translation of \(\triangle ABC\). 

7. Show that \(ABCD \cong KLMJ\) using translations.
   
   The coordinates of \(ABCD\) are \(A(-5,5)\), \(B(-2,4)\), \(C(-4,1)\), and \(D(-8,2)\).
   
   The coordinates of \(KLMJ\) are \(K(3,1)\), \(L(6,0)\), \(M(4,-3)\), and  \(J(0,-2)\).

   
8. The following equations all represent linear relationships.
   Group A
   \(y=2x\)
   \(y=2x-2\)
   \(y=2x+5\)

   Group B
   \(y=x\)
   \(y=x-2\)
   \(y=x+5\)
   1. Using a table of values, graph each linear relationship. Graph the three equations from the same group on the same graph. 
   2. What do you notice about the rate of change of the linear relationships in each group?
   3. Using the language of translations, describe how the graphs of the linear relationships within each group are related.
   4. Consider the linear relationship \(y=4x\) on the following graph:
      
      Without using an equation, graph two more linear relationships that can be obtained by translating the relationship \(y=4x\).