Question

the square ( mnop ) is a dilation of the square ( mnop ). what is the scale factor of the dilation?
simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of a side of the original square

The original square \(MNOP\). Let's find the length of \(NO\). The \(x -\)coordinates of \(N(- 5,4)\) and \(O(3,4)\). Using the distance formula for horizontal distance \(d=\vert x_2 - x_1\vert\). Here \(d=\vert3-(-5)\vert=\vert3 + 5\vert = 8\) units.

Step2: Find the length of a side of the dilated square

The dilated square \(M'N'O'P'\). Let's find the length of \(N'O'\). The \(x -\)coordinates of \(N'(-10,8)\) and \(O'(8,8)\). Using the distance formula for horizontal distance \(d=\vert x_2 - x_1\vert\). Here \(d=\vert8-(-10)\vert=\vert8 + 10\vert=18\) units.

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side of dilated figure}}{\text{length of side of original figure}}\). But wait, let's use another approach. If we consider the ratio of the \(y -\)coordinates (since dilation is uniform). For point \(N(-5,4)\) and \(N'(-10,8)\). The scale factor \(k=\frac{y_{N'}}{y_N}\) (or \(\frac{x_{N'}}{x_N}\) as the dilation is centered at the origin, assume). \(\frac{8}{4}=2\)

Answer:

\(2\)