Question

1. quadrilateral efgh is dilated to create quadrilateral efgh using the origin as the center of dilation. which one of the following statements is true? a quadrilateral efgh is dilated by a scale factor of 1/4 to create quadrilateral efgh. b quadrilateral efgh is dilated by a scale factor of 1/3 to create quadrilateral efgh. c quadrilateral efgh is dilated by a scale factor of 1/2 to create quadrilateral efgh. d quadrilateral efgh is dilated by a scale factor of 1/1 to create quadrilateral efgh

Explanation:

Step1: Select a point

Let's take point E(1, 12) and its image E'(2, 6).

Step2: Calculate scale - factor for x - coordinates

The scale - factor \(k_x=\frac{x_{E'}}{x_E}=\frac{2}{1} = 2\) (this is wrong way, we should consider the ratio of the new to the old in the context of dilation). Let's use the correct approach. If we consider the ratio of the coordinates of the dilated figure to the original figure in terms of distance from the origin. For a point \((x,y)\) dilated to \((x',y')\) with the origin as the center of dilation, the scale - factor \(k=\frac{\text{distance of }(x',y')\text{ from origin}}{\text{distance of }(x,y)\text{ from origin}}\). Another way is to look at the ratio of corresponding side lengths. Let's take the horizontal distance between two points. For example, if we consider the horizontal distance between two points on the quadrilateral. Let's say the horizontal distance between two points on EFGH and E'F'G'H'. Suppose we consider the horizontal distance between two vertices of the quadrilaterals. Let's take the x - coordinates of two corresponding vertices. Let's take point G(14,4) and G'(7,2).
The scale - factor \(k=\frac{x_{G'}}{x_G}=\frac{7}{14}=\frac{1}{2}\). We can also check with the y - coordinates. For point G, \(y = 4\) and for G', \(y'=2\), and \(\frac{y_{G'}}{y_G}=\frac{2}{4}=\frac{1}{2}\).

Answer:

C. Quadrilateral EFGH is dilated by a scale factor of \(\frac{1}{2}\) to create quadrilateral E'F'G'H'