An illustration shows an 11 by 10 grid that has 3 rectangles on it. Figure 1 covers the 2nd row in the 3rd through 6th columns, for a total of 4 units. Figure 2 covers the 2nd to 9th rows in the 9th and 10th columns, for a total of 16 units. Figure 3 covers the 4th to the 7th rows in the 4th and 5th columns, for a total of 8 units.

Question

Which combination of figures are similar and why?

Figure 1 and Figure 3; one figure is larger than the other.
Figure 1 and Figure 3; one figure is larger than the other.

Figure 1 and Figure 2; the side lengths are proportional and corresponding angles are equal.
Figure 1 and Figure 2; the side lengths are proportional and corresponding angles are equal.

Figure 1, Figure 2, and Figure 3; all figures are rectangles.
Figure 1, Figure 2, and Figure 3; all figures are rectangles.

Figure 2 and Figure 3; the figures are both vertical.

Answer 1

To determine which combination of figures are similar, we need to analyze the properties of the rectangles.

Figure 1 covers a row, specifically the 2nd row from the 3rd to the 6th columns. Thus, it has dimensions:

• Width: 4 units (columns 3 to 6)
• Height: 1 unit (only one row)

Figure 2 covers multiple rows from the 2nd to the 9th rows in the 9th and 10th columns, giving it dimensions:

• Width: 2 units (columns 9 and 10)
• Height: 8 units (rows 2 to 9)

Figure 3 covers rows from the 4th to the 7th in the 4th and 5th columns. Its dimensions are:

• Width: 2 units (columns 4 and 5)
• Height: 4 units (rows 4 to 7)

Now, we can check the similarity based on the dimensions:

1. Figure 1 and Figure 2:

   • Figure 1: 4 units by 1 unit
   • Figure 2: 2 units by 8 units The side lengths are not proportional (4:1 vs. 2:8).

2. Figure 1 and Figure 3:

   • Figure 1: 4 units by 1 unit
   • Figure 3: 2 units by 4 units The side lengths are not proportional either (4:1 vs. 2:4).

3. Figure 2 and Figure 3:

   • Figure 2: 2 units by 8 units
   • Figure 3: 2 units by 4 units Here, both figures have 2 units as their width. However, the heights aren’t proportional (8 units for Figure 2 and 4 units for Figure 3) which indicates they are not similar.

4. All Figures:

   • All figures are rectangles, but that does not imply they are similar (as they do not have proportional sides).

Based on this analysis, none of the figures are similar to each other based on the criteria of proportional side lengths and equal corresponding angles. Therefore the best choice would lean towards acknowledging that while they share the property of being rectangles, they do not demonstrate similarity since similarity requires more than just being rectangles; it requires proportional dimensions.

Ultimately, the most accurate option given your choices would be:

• Figure 1, Figure 2, and Figure 3; all figures are rectangles.

However, it’s important to note that this does not signify similarity in shape.