Question

what would the center of dilation be for the following object with a at the point (-3,3) and b at the point (-1.5,3)? (0,2) (-1,2) (-1,1) not enough information

Explanation:

Step1: Recall dilation formula

For a dilation with center \((h,k)\), the formula for a point \((x,y)\) and its image \((x',y')\) is \(x'=h + k(x - h)\) and \(y'=k+(y - k)\). Here the center of dilation is \((- 3,3)\) and we know \(A(-3,3)\) maps to \(A'(-3,3)\) (since the center of dilation doesn't move under dilation). Let the scale - factor be \(k\). To find the image of \(B\) under dilation, we first note that the distance between the center of dilation \((-3,3)\) and the new point \(A'(-3,3)\) is used to find the transformation for other points. The transformation of a point \((x,y)\) to \((x',y')\) with center of dilation \((h,k)\) can also be thought of in terms of the vector from the center of dilation. The vector from \((-3,3)\) to \(A\) is \((0,0)\) and to the new \(A'\) is \((0,0)\). Let's assume the scale - factor \(k=\frac{-1.5+3}{-3 + 3}=\frac{1.5}{0}\) (this is wrong approach, we use the property of similar - triangles or vector approach). If we consider the ratio of the distances from the center of dilation. The distance from the center of dilation \((-3,3)\) to \(A(-3,3)\) and to \(A'(-3,3)\) gives us the idea. For a point \(B\), if we consider the relative position with respect to the center of dilation. The vector from the center of dilation \((-3,3)\) to a general point \((x,y)\) is \(\vec{v}=(x + 3,y - 3)\). After dilation, the new vector \(\vec{v}'\) is related to \(\vec{v}\) by the scale - factor. Since \(A(-3,3)\) maps to \(A'(-3,3)\), we can use the fact that if we consider the coordinates of \(B\) and the center of dilation \((-3,3)\) and the new point \(A'(-3,3)\). Let's assume the transformation rule \((x,y)\to(x',y')\) where \(x'=-3+(x + 3)\times\frac{-1.5+3}{-3+3}\) and \(y'=3+(y - 3)\times\frac{-1.5+3}{-3+3}\). A simpler way is to consider the ratio of the \(x\) and \(y\) displacements. The displacement of \(A\) from the center of dilation is \(0\) in both \(x\) and \(y\) directions. For a point \(B\), if we consider the transformation based on the fact that the center of dilation \((-3,3)\) and the new position of \(A\) (which is the same as the center of dilation). The transformation of a point \((x,y)\) with center of dilation \((h,k)\) is given by \(x'=h+(x - h)r\) and \(y'=k+(y - k)r\) where \(r\) is the scale - factor. Since \(A(-3,3)\) maps to \(A'(-3,3)\), we can find the position of \(B'\) as follows: The \(x\) - coordinate of \(B'\): Let the center of dilation \(C(-3,3)\), for \(B\) (assume we know the relative position from the center of dilation). The \(x\) - coordinate of \(B'\): \(x=-3+(x_B + 3)\times\frac{-1.5+3}{-3+3}\), but we can also use the fact that if we consider the linear relationship. If we assume the center of dilation \((-3,3)\) and we know \(A\) and \(A'\) are the same. The \(x\) - coordinate of \(B'\): \(x=-1\) (by considering the ratio of the \(x\) - displacements from the center of dilation). The \(y\) - coordinate of \(B'\): \(y = 1\) (similarly, by considering the \(y\) - displacement from the center of dilation). So the coordinates of \(B'\) are \((-1,1)\).

Answer:

(-1,1)