Question

11.
original figure (square with side 2)
scale factor: 4
area of the original figure: 4
area of the scale drawing: 100 (handwritten)
12.
original figure (square with side 2)
scale factor: 100
area of the original figure: 4 (handwritten)
area of the scale drawing: 400 (handwritten)

1. do you think your conjecture will be true for scaled figures that are not squares? explain your thinking.

Explanation:

For non - square scaled figures, the relationship between the area of the original figure and the area of the scaled figure is based on the concept of similar figures. When a figure is scaled by a scale factor \( k \), if the figure is similar to the original (which is the case for scaled figures, as scaling preserves the shape, i.e., the angles remain the same and the sides are in proportion), the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding side lengths (the scale factor).

For example, consider a rectangle with length \( l \) and width \( w \). The area of the original rectangle \( A_{original}=l\times w \). If we scale it by a scale factor \( k \), the new length \( l' = k\times l \) and the new width \( w'=k\times w \). The area of the scaled rectangle \( A_{scaled}=l'\times w'=(k\times l)\times(k\times w)=k^{2}\times(l\times w) = k^{2}\times A_{original} \).

Another example is a triangle with base \( b \) and height \( h \). The area of the original triangle \( A_{original}=\frac{1}{2}\times b\times h \). When scaled by a factor \( k \), the new base \( b' = k\times b \) and the new height \( h'=k\times h \). The area of the scaled triangle \( A_{scaled}=\frac{1}{2}\times b'\times h'=\frac{1}{2}\times(k\times b)\times(k\times h)=k^{2}\times(\frac{1}{2}\times b\times h)=k^{2}\times A_{original} \).

So, regardless of whether the figure is a square, a rectangle, a triangle, or any other polygon (as long as it is a scaled (similar) figure), the area of the scaled figure is the square of the scale factor times the area of the original figure. So the conjecture (that the area of the scaled figure is the square of the scale factor times the area of the original figure) should hold for non - square scaled figures.

Answer:

Yes, the conjecture will be true for scaled figures that are not squares. This is because for any similar figures (scaled figures have the same shape, so they are similar), the ratio of their areas is equal to the square of the ratio of their corresponding side lengths (the scale factor). For example, for a rectangle with length \( l \) and width \( w \), original area \( A = lw \). If scaled by factor \( k \), new length \( kl \), new width \( kw \), new area \( (kl)(kw)=k^{2}lw = k^{2}A \). For a triangle with base \( b \) and height \( h \), original area \( A=\frac{1}{2}bh \). Scaled by \( k \), new base \( kb \), new height \( kh \), new area \( \frac{1}{2}(kb)(kh)=k^{2}(\frac{1}{2}bh)=k^{2}A \). So the relationship (area of scaled figure = scale factor²× area of original figure) holds for non - square scaled (similar) figures.