Exercises

1. Square \(EFGH\) is obtained from square \(ABCD\) using a dilatation with centre \((0,0)\). What is the scale factor of the dilatation? Explain.
   We have points \(A(-1,1)\), \(B(1,1)\), \(C(1,-1)\), and \(D(-1,-1)\).  We also have points \(E(-2,2)\), \(F(2,2)\), \(G(2,-2)\), and \(H(-2,-2)\).
   
2. Use a ruler to construct a similar polygon with the given scale factor and indicated centre of dilatation. 
   1. Scale factor of \(\dfrac{1}{2}\).
      
   2. Scale factor of \(3\).
      
3. 1. Draw a parallelogram that is similar to parallelogram \(STUV\) by a scale factor of \(2\). Use \((0,3)\) as the centre of dilatation. 
      We have points \(S(-4,5)\), \(T(0,5)\), \(U(3,0)\), and \(V(-1,0)\).
   2. Draw a parallelogram that is similar to parallelogram \(LMNO\) by a scale factor of \(\dfrac{1}{2}\). Use \((0,0)\) as the centre of dilatation. 
      We have points \(L(1,-3)\), \(M(4,0)\), \(N(8,0)\), and \(O(5,-3)\).
      
4. Mark each statement as true or false. Explain your answers.
   1. A dilatation of a quadrilateral must be another quadrilateral.
   2. A dilatation of a rectangle cannot be a square.
   3. A dilatation of a right-angled triangle must be another right-angled triangle.
   4. If a polygon is dilated with a scale factor of \(5\), then the area of the image will be larger than the area of the original polygon.
   5. If a polygon with vertex \(A(2,3)\) is dilated with a scale factor of \(2\), then the \(x\)-coordinate of the image of \(A\) is greater than \(2\).
   6. If rectangle \(ABCD\) has vertex \(A(1,1)\), then you can find a dilatation of \(ABCD\) under which the image of point \(A\) is \(A'(2,2)\).
   7. If rectangle \(ABCD\) has vertex \(A(1,1)\), then you can find a dilatation of \(ABCD\) under which the image of point \(A\) is \(A'(1,1)\).
5. Is \(\triangle AED\) a dilatation of \(\triangle ABC\)? If so, what is the scale factor and centre of the dilatation?
   
6. Quadrilateral \(ABCD\) is a dilatation of \(PQRS\). What is the scale factor and the centre of the dilatation?
   We have points \(A(3,0)\), \(B(6,0)\), \(C(3,-6)\), and \(D(-6,-6)\). 
   We also have points \(P(1,0)\), \(Q(2,0)\), \(R(1,-2)\), and \(S(-2,-2)\).
   
7. In this question, we will explore what happens to the vertices of a quadrilateral under a dilatation with a centre of \((0,0).\)
   1. For each part below, the vertices of a quadrilateral \(ABCD\) are given. Determine the vertices of the image of the quadrilateral, \(A'B'C'D'\), after a dilatation with centre \((0,0)\) and scale factor of \(2\).
      1. We have points \(A(-1,2)\), \(B(1,2)\), \(C(1,-2)\), and \(D(-1,-2)\).
         
      2. We have points \(A(0,1)\), \(B(4,1)\), \(C(4,-2)\), and \(D(0,-2)\).
         
      3. We have points \(A(0,3)\), \(B(3,0)\), \(C(2,-2)\), and \(D(-4,-1)\).
   2. Predict the answers to part a) if the centre of the dilatation is again \((0,0)\) but instead the scale factor is \(3\).
   3. Suppose that one of the vertices of a quadrilateral is \(A(x,y)\). If a dilatation with centre \((0,0)\) and scale factor \(s\) is performed on this quadrilateral, then what are the coordinates of the image of vertex \(A\)? 
8. 1. Suppose that one of the vertices of a polygon is \(P(x,y)\). If a dilatation with centre \((1,2)\) and scale factor \(3\) is performed on this polygon, then what are the coordinates of the image of vertex \(P\)? 
   2. Suppose that one of the vertices of a polygon is \(Q(x,y)\). If a dilatation with centre \((1,2)\) and scale factor \(s\) is performed on this polygon, then what are the coordinates of the image of vertex \(Q\)?