Question

1. triangle def is the image that resulted from a dilation of △abc using the origin as the center of dilation. what scale factor was used to dilate △abc? the corresponding angles are all original side length : new side length is the coordinates of a ( ) the coordinates of d ( ) scale factor = \\(\frac{new}{original}=\frac{}{}=\\) the area of △abc is 4 units² the area of △def is 4 x = unit² the algebraic representation is ( )

Explanation:

Step1: Find scale factor from coordinates

For \(A(2,3)\) and \(D(4,6)\), \(k=\frac{4}{2}=2\)

Step2: Find area of \(\triangle DEF\)

Since \(k = 2\), area of \(\triangle DEF=4\times k^{2}=4\times4 = 16\)

Step3: Determine algebraic representation

With \(k = 2\), \((x,y)\to(2x,2y)\)

Step4: Identify side - length ratio

If original side \(=2\) and new side \( = 4\), ratio is \(1:2\)

Step5: Recall angle property

Similar triangles have equal corresponding angles.

Answer:

Let's assume we can identify the coordinates and side - lengths from the graph. Suppose the coordinates of \(A=(2,3)\), \(D=(4,6)\).
The scale factor \(k\) for dilation with the origin as the center of dilation is given by the ratio of the coordinates of the corresponding points. For a point \((x,y)\) and its dilated image \((x',y')\), \(k=\frac{x'}{x}=\frac{y'}{y}\). Here, for point \(A(2,3)\) and \(D(4,6)\), \(k = \frac{4}{2}=\frac{6}{3}=2\).
If the scale factor of dilation is \(k\), the ratio of the areas of two similar figures (the original and the dilated one) is \(k^{2}\). Since \(k = 2\), if the area of \(\triangle ABC\) is \(4\) square units, the area of \(\triangle DEF\) is \(4\times k^{2}=4\times4 = 16\) square units.
The algebraic representation of a dilation with the origin as the center of dilation and scale factor \(k\) is \((x,y)\to(kx,ky)\), so it is \((x,y)\to(2x,2y)\).

The corresponding angles of similar triangles (a dilated triangle and the original triangle are similar) are equal.
If we assume we can measure side - lengths, say an original side - length of \(\triangle ABC\) is \(a\) and the corresponding side - length of \(\triangle DEF\) is \(b\), and we find \(a = 2\) and \(b=4\), then the ratio of the original side - length to the new side - length is \(1:2\).