Question

1. square j k l m is dilated with the origin as the center of dilation to create square j k l m. identify the scale factor and rule that represents the dilation. scale factor: algebraic rule: (x, y) → (2x, 2y) enlargement or reduction? enlargement

Explanation:

Step1: Understand the concept of scale factor

The scale factor \(k\) in a dilation \((x,y)\to(kx,ky)\) can be found by comparing the lengths of corresponding sides of the original and dilated figure. Let's assume a side length of the original square \(JKLM\). For example, if we consider the distance from the center (origin) to a vertex. Suppose a vertex of \(JKLM\) is at \((5,0)\) and the corresponding vertex of \(J'K'L'M'\) is at \((2.5,0)\)

Step2: Calculate the scale factor

The formula for scale factor \(k=\frac{\text{length of side of dilated figure}}{\text{length of side of original figure}}\). If \(x\) (or \(y\)) - coordinate of a vertex of the original square is \(x_1\) and of the dilated square is \(x_2\), then \(k = \frac{x_2}{x_1}\). Let \(x_1 = 5\) (from \(J=(5,0)\)) and \(x_2=2.5\) (from \(J'=(2.5,0)\)). Then \(k=\frac{2.5}{5}=\frac{1}{2}\)

Answer:

Scale Factor: \(\frac{1}{2}\), Enlargement or Reduction? Reduction, Algebraic Rule: \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)