Question

lewis draws a rectangle with dimensions 9 units by 5 units, then dilates it by a scale factor of 3. how many times greater is the area of the dilated rectangle than the area of the original rectangle? (1 point) 45 times greater 27 times greater 3 times greater 9 times greater

Explanation:

Step1: Find original area

The area of a rectangle is given by \( A = l \times w \). For the original rectangle with length \( l = 9 \) and width \( w = 5 \), the area \( A_{original}=9\times5 = 45 \) square units.

Step2: Find dilated dimensions

When dilating by a scale factor of \( 3 \), the new length \( l_{new}=9\times3 = 27 \) and new width \( w_{new}=5\times3 = 15 \).

Step3: Find dilated area

The area of the dilated rectangle \( A_{dilated}=27\times15 = 405 \) square units.

Step4: Find the ratio

To find how many times greater the dilated area is, we calculate the ratio \( \frac{A_{dilated}}{A_{original}}=\frac{405}{45}=9 \). Alternatively, we can use the property that when a figure is dilated by a scale factor \( k \), the area is dilated by \( k^{2} \). Here, \( k = 3 \), so the area scale factor is \( 3^{2}=9 \).

Answer:

9 times greater (corresponding to the option: 9 times greater)