Question

points a and b represent points on a safety pin. points a and b represent points on the image of the safety pin after a dilation centered at the origin. what is the scale factor of the dilation?

Explanation:

Step1: Recall the formula for scale factor

The formula for the scale factor \(k\) of a dilation centered at the origin is \(k=\frac{x_{image}}{x_{pre - image}}\) (or \(\frac{y_{image}}{y_{pre - image}}\)) for a point \((x,y)\) and its image \((x',y')\) after dilation.

Step2: Use point \(A\) and \(A'\) to find the scale factor

For point \(A(4,4)\) and its image \(A'(6,6)\). Using the \(x\) - coordinates: \(k = \frac{x_{A'}}{x_{A}}\). Substitute \(x_{A}=4\) and \(x_{A'}=6\) into the formula. \(k=\frac{6}{4}=\frac{3}{2}\). Using the \(y\) - coordinates: \(k=\frac{y_{A'}}{y_{A}}\), substitute \(y_{A}=4\) and \(y_{A'}=6\) into the formula. \(k = \frac{6}{4}=\frac{3}{2}\).

Step3: Verify with point \(B\) and \(B'\)

For point \(B(6,2)\) and its image \(B'(9,3)\). Using the \(x\) - coordinates: \(k=\frac{x_{B'}}{x_{B}}\), substitute \(x_{B}=6\) and \(x_{B'}=9\) into the formula. \(k=\frac{9}{6}=\frac{3}{2}\). Using the \(y\) - coordinates: \(k=\frac{y_{B'}}{y_{B}}\), substitute \(y_{B}=2\) and \(y_{B'}=3\) into the formula. \(k=\frac{3}{2}\).

Answer:

\(\frac{3}{2}\)