Recall that in a scale drawing, all the dimensions maintain the same ratio with their corresponding actual dimensions.
100 times the area of the sketch
Practice makes perfect
The dimensions of the sketch of a rug are 4 inches by 7.5 inches. The actual rug is 10 times bigger, so the scale is 1:10. The problem asks how many times (larger) is the area of the actual rug compared to the area of the sketch.
Which Figures Are Similar?
Since the rug in the sketch is drawn to scale, the sketched rug and the actual rug will keep the same proportions between corresponding sides. For this reason, they will be similar.
Using Proportions to Find the Dimensions of the Actual Rug
As the scale is 1:10, for each inch in any of the sketched rug's dimensions the real rug has 10 inches. For this reason, the dimensions of the actual rug will be those of the sketch multiplied by 10.
Sketch (inches)
Actual (inches)
Width
4
4 * 10= 40
Length
7.5
7.5 * 10 = 75
Finding and Relating the Areas
The area of the rectangular rug in the sketch will be found using the formula for the area of a rectangle.
The area of the actual rug will be found in the same manner.
\begin{aligned}
A_\text{Rug} \ &= \ \text{Rug Width} \times \text{Rug Length}
\end{aligned}
Using the scale factor of 1:10, we can find the actual area of the rug given the known dimensions of the sketch.
