Question

1. here are two similar polygons. measure the side lengths and angles of each polygon. what do you notice? 3. each figure shows a pair of similar triangles, one contained in the other. for each pair, describe a point and a scale factor to use for a dilation moving the larger triangle to the smaller one. use a measurement tool to find the scale factor.

Explanation:

Step 1: For problem 2

• Measure side - lengths using a ruler and angles using a protractor. For similar polygons, corresponding angles are equal. Let's assume polygon \(ABCD\) and polygon \(EFGH\) are similar.
• When measuring side - lengths, we will find that the ratios of corresponding side - lengths are equal. For example, if \(AB\) and \(EF\) are corresponding sides, \(\frac{AB}{EF}\) is a constant (the scale factor). And when measuring angles, \(\angle A=\angle E\), \(\angle B=\angle F\), \(\angle C=\angle G\), \(\angle D=\angle H\).

Step 2: For problem 3

• To find the center of dilation (the point), we look for the point from which the dilation is occurring. If the lines connecting corresponding vertices of the two similar triangles intersect at a single point, that point is the center of dilation.
• To find the scale factor \(k\), we use the formula \(k=\frac{\text{length of a side in the smaller triangle}}{\text{length of the corresponding side in the larger triangle}}\). For example, in \(\triangle A'B'C'\) and \(\triangle ABC\), if we measure \(A'B'\) and \(AB\), \(k = \frac{A'B'}{AB}\).

Answer:

For problem 2: Corresponding angles of similar polygons are equal and ratios of corresponding side - lengths are equal.
For problem 3: First, find the center of dilation by finding the intersection - point of lines connecting corresponding vertices. Then, use the ratio of corresponding side - lengths to find the scale factor \(k=\frac{\text{side length in smaller triangle}}{\text{side length in larger triangle}}\).