Question

11
original figure (2×2 square on grid), scale factor 4, area of the original figure, area of the scale drawing
12
original figure (2×2 square on grid), scale factor 100, area of the original figure, area of the scale drawing

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

Explanation:

Problem 11

Step1: Find area of original square

The original figure is a square with side length \( s = 2 \). The formula for the area of a square is \( A = s^2 \). So, \( A_{\text{original}} = 2^2 = 4 \).

Step2: Find area of scale drawing

When a figure is scaled by a scale factor \( k \), the area of the scaled figure is \( A_{\text{scaled}} = k^2\times A_{\text{original}} \). Here, scale factor \( k = 4 \). So, \( A_{\text{scaled}} = 4^2\times4 = 16\times4 = 64 \).

Step1: Find area of original square

The original figure is a square with side length \( s = 2 \). Using the area formula for a square \( A = s^2 \), we get \( A_{\text{original}} = 2^2 = 4 \).

Step2: Find area of scale drawing

With scale factor \( k = 100 \), the area of the scaled figure is \( A_{\text{scaled}} = k^2\times A_{\text{original}} \). So, \( A_{\text{scaled}} = 100^2\times4 = 10000\times4 = 40000 \).

When a figure (not just a square) is scaled by a scale factor \( k \), each linear dimension (length, width, height, etc.) is multiplied by \( k \). For any two - dimensional figure, the area is a product of two linear dimensions (e.g., for a rectangle, \( A = l\times w \); for a triangle, \( A=\frac{1}{2}bh \); for a circle, \( A=\pi r^2 \) where \( r \) is a linear dimension). If the linear dimensions are scaled by \( k \), then for a rectangle with original length \( l \) and width \( w \), the new length is \( kl \) and new width is \( kw \), so new area \( A'=(kl)\times(kw)=k^{2}lw = k^{2}A \). For a triangle with original base \( b \) and height \( h \), new base \( kb \) and new height \( kh \), new area \( A'=\frac{1}{2}(kb)(kh)=\frac{1}{2}k^{2}bh = k^{2}A \). For a circle with original radius \( r \), new radius \( kr \), new area \( A'=\pi(kr)^{2}=k^{2}\pi r^{2}=k^{2}A \). So the relationship between the area of the original figure and the area of the scaled figure (where the scale factor is applied to linear dimensions) holds for non - square figures as well, because area is a two - dimensional measurement and scaling linear dimensions by \( k \) will scale the area by \( k^{2} \).

Answer:

• Scale Factor: 4
• Area of Original Figure: \( 4 \)
• Area of Scale Drawing: \( 64 \)

Problem 12