Question

2 multiple choice 5 points
△abc is the original figure and △abc represents its dilation image. fill in the blanks:
△abc is a dilation of △abc by a factor of ____ about the point ____
2; (2,2)
2; (1,2)
3; (4,2)
3; (1,2)

Explanation:

Step1: Recall the formula for dilation

The formula for dilation about a point \((h,k)\) with scale factor \(s\) is \((x,y)\to s(x - h)+h,s(y - k)+k\).

Step2: Identify the center of dilation

Let's assume the center of dilation is \((h,k)\). For a dilation, if we consider a point \(B\) and its image \(B'\). Looking at the coordinates (by counting the grid - squares), we can observe that the distance from the center of dilation to the original point and its image is related by the scale factor. If we assume the center of dilation is \((1,2)\)

Step3: Calculate the scale factor

Take a point \(A\). Suppose the coordinates of \(A\) (by counting grid - squares) is \((2,3)\) and the coordinates of \(A'\) is \((3,5)\). Using the dilation formula \(x'=s(x - h)+h\) and \(y'=s(y - k)+k\) with \((h,k)=(1,2)\). For the \(x\) - coordinate: \(x'=s(x - 1)+1\), substituting \(x = 2\) and \(x'=3\), we get \(3=s(2 - 1)+1\), which simplifies to \(s=2\). For the \(y\) - coordinate: \(y'=s(y - 2)+2\), substituting \(y = 3\) and \(y'=5\), we get \(5=s(3 - 2)+2\), which also gives \(s = 2\)

Answer:

2; \((1,2)\)