Question

lesson 2 – practice problems 1. the second h - shaped polygon is a scaled copy of the first. a. show one pair of corresponding points and two pairs of corresponding sides of the original polygon and its copy. consider using colored pencils to highlight corresponding parts or labeling some of the vertices. b. what scale factor takes the original polygon to its smaller copy? explain or show your reasoning.

Explanation:

Part a (Visual Identification, No Calculation Here)

To show corresponding points and sides:

1. Identify the Original and Copy: The larger H - shaped figure is the original, the smaller one is the copy.
2. Label Vertices: Let's label the top - left vertex of the original as \(A\) and the top - left vertex of the copy as \(A'\). Similarly, label the top - right vertex of the original as \(B\) and the top - right vertex of the copy as \(B'\). For sides, the top horizontal side of the original (between \(A\) and \(B\)) corresponds to the top horizontal side of the copy (between \(A'\) and \(B'\)). Another pair: the bottom - left vertex of the original \(C\) and bottom - left vertex of the copy \(C'\), with the bottom horizontal side between \(C\) and its right counterpart in the original corresponding to the bottom horizontal side between \(C'\) and its right counterpart in the copy. (This is a visual matching; you can use colored pencils to highlight these corresponding vertices and sides.)

Part b (Calculating Scale Factor)

Step 1: Choose a Corresponding Side

Pick a side from the original polygon and its corresponding side from the copy. For example, look at the length of the top horizontal side of the original H - shape. Let's assume each grid square has a side length of 1 unit. If the top horizontal side of the original spans, say, 6 grid units (by counting the number of squares it covers horizontally), and the top horizontal side of the copy spans 2 grid units.

Step 2: Calculate Scale Factor

The scale factor \(k\) from the original to the copy is given by the ratio of the length of the side in the copy to the length of the corresponding side in the original. So \(k=\frac{\text{Length of side in copy}}{\text{Length of side in original}}\). If the original side length \(L_{original} = 6\) and the copy side length \(L_{copy}=2\), then \(k = \frac{2}{6}=\frac{1}{3}\). (We can verify with other sides: if the vertical "stem" of the original is, say, 6 units and the vertical stem of the copy is 2 units, the ratio is also \(\frac{2}{6}=\frac{1}{3}\), confirming the scale factor.)

Answer:

a. (Visual: Label corresponding vertices (e.g., top - left, top - right, bottom - left, bottom - right of original and copy) and highlight corresponding sides (e.g., top horizontal, bottom horizontal, vertical stem) with colored pencils.)
b. The scale factor is \(\frac{1}{3}\). We find the length of a corresponding side in the original and the copy (by counting grid units), then take the ratio of the copy's side length to the original's side length. For example, if a horizontal side of the original is 6 units and the copy's is 2 units, \(\frac{2}{6}=\frac{1}{3}\), and this ratio holds for other corresponding sides.