Question

lesson 6: practice problems

1. each diagram has a pair of figures, one larger than the other. for each pair, show that the two figures are similar by identifying a sequence of translations, rotations, reflections, and dilations that takes the smaller figure to the larger one.

Explanation:

Step1: Analyze the first - grid - based figure

For the first pair of figures on the rectangular grid (triangle \(ABC\) and triangle \(DEF\)), first, we note that there is no rotation or reflection. The transformation from \(\triangle ABC\) to \(\triangle DEF\) is a dilation centered at the origin \((0,0)\) and a translation. The dilation factor \(k = 2\) since the side - lengths of \(\triangle DEF\) are twice those of \(\triangle ABC\). After dilation, we translate the dilated triangle to the right and up to match \(\triangle DEF\).

Step2: Analyze the second - polar - grid - based figure

For the second pair of figures on the polar grid (quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\)), we first identify the center of dilation as point \(A\). The dilation factor \(k\) can be determined by comparing the distances of corresponding points from the center of dilation. Then, we may need to perform a rotation (if the orientation of the figures is different) around point \(A\) to map the dilated quadrilateral to \(A'B'C'D'\).

For the first pair of figures (rectangular grid):

1. Dilation: The scale factor \(k = 2\) centered at the origin \((0,0)\). If a point \((x,y)\) in \(\triangle ABC\), after dilation, the new point \((x',y')=(2x,2y)\).
2. Translation: After dilation, translate the dilated triangle so that it coincides with \(\triangle DEF\).

For the second pair of figures (polar grid):

1. Dilation: Determine the scale factor \(k\) by comparing the distances of corresponding points from the center of dilation \(A\). For example, if the distance of a point \(B\) from \(A\) is \(r_1\) and the distance of \(B'\) from \(A\) is \(r_2\), then \(k=\frac{r_2}{r_1}\).
2. Rotation: Rotate the dilated quadrilateral around point \(A\) to match the orientation of \(A'B'C'D'\).

Answer:

For the first pair of figures on the rectangular grid: Dilation with a scale factor of 2 centered at the origin followed by a translation. For the second pair of figures on the polar grid: Dilation centered at point \(A\) followed by a rotation around point \(A\).