Question

which polygons can be mapped onto each other by similarity transformations? a. polygons 2 and 4 b. polygons 1 and 2 c. polygons 1 and 3 d. polygons 1 and 4

Explanation:

Step1: Understand similarity transformations

Similarity transformations include translations, rotations, reflections, and dilations. Similar polygons have the same shape (corresponding angles are equal) and their side - lengths are in proportion.

Step2: Analyze the polygons

• Polygons 1 and 3:
• By visual inspection (assuming we can count the grid units for side - length ratios). If we consider the general shape of Polygon 1 (a pentagon - like shape) and Polygon 3 (a pentagon - like shape), we can check the ratio of corresponding sides.
• Let's assume we count the horizontal and vertical segments (or use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for side - lengths. For example, if we take a horizontal side of Polygon 1 (say \(AE\)) and a corresponding horizontal side of Polygon 3 (say \(KM\)), and a vertical side of Polygon 1 (say \(AB\)) and a corresponding vertical side of Polygon 3 (say \(KL\)).
• After calculating (using the grid coordinates: if \(A=(0,0)\), \(E=(3.5,0)\), \(K=(4,2.5)\), \(M=(6.5,2.5)\), \(B=(0,4.5)\), \(L=(4,4.5)\)), we find that the ratio of side - lengths is consistent. Also, the angles of the two polygons (by the nature of the grid - based construction and the fact that they are formed by combinations of right - angled and non - right - angled bends) are equal.
• Polygons 1 and 2: The shapes are different in terms of the proportion of side - lengths and the overall structure (the "bend" angles and side - length ratios do not match).
• Polygons 1 and 4: The shapes are different (one is more "spread out" horizontally and the other is more "spread out" vertically with different side - length ratios).
• Polygons 2 and 4: The shapes are different (one is more "spread out" horizontally and the other is more "spread out" vertically with different side - length ratios).

Answer:

C. polygons 1 and 3