The smaller of two similar rectangles has dimensions 4 and 6. Find the dimensions of the larger rectangle if the ratio of the perimeters is $2$ to $3$.
A. $22 / 3$ by 4
B. 6 by 9
C. 8 by 18
D. 12 by 12
Answer (1)
The perimeters of similar rectangles are in the same ratio as their sides.
Calculate the perimeter of the smaller rectangle: P 1 = 2 ( 4 + 6 ) = 20 .
Find the perimeter of the larger rectangle using the given ratio: P 2 = 2 3 P 1 = 30 .
Determine the dimensions of the larger rectangle by scaling the sides of the smaller rectangle by 2 3 : 6 × 2 3 = 9 and 4 × 2 3 = 6 . Thus, the dimensions are 6 by 9 .
Explanation
Problem Analysis We are given two similar rectangles. The smaller rectangle has dimensions 4 and 6. The ratio of the perimeters of the smaller to the larger rectangle is 2:3. We need to find the dimensions of the larger rectangle.
Define Variables and Similarity Let the dimensions of the smaller rectangle be l 1 = 6 and w 1 = 4 . Let the dimensions of the larger rectangle be l 2 and w 2 . Since the rectangles are similar, the ratio of corresponding sides is constant: l 2 l 1 = w 2 w 1 = k , where k is the similarity ratio.
Calculate Perimeters The perimeter of the smaller rectangle is P 1 = 2 ( l 1 + w 1 ) = 2 ( 6 + 4 ) = 2 ( 10 ) = 20 . The perimeter of the larger rectangle is P 2 = 2 ( l 2 + w 2 ) .
Use Perimeter Ratio The ratio of the perimeters is given as P 2 P 1 = 3 2 . Therefore, P 2 = 2 3 P 1 = 2 3 ( 20 ) = 30 .
Find Dimensions of Larger Rectangle So, 2 ( l 2 + w 2 ) = 30 , which means l 2 + w 2 = 15 . Since l 2 l 1 = w 2 w 1 = 3 2 , we have l 2 = 2 3 l 1 = 2 3 ( 6 ) = 9 and w 2 = 2 3 w 1 = 2 3 ( 4 ) = 6 .
Verification Check if l 2 + w 2 = 15 : 9 + 6 = 15 . This confirms the dimensions of the larger rectangle are 9 and 6.
Final Answer The dimensions of the larger rectangle are 6 by 9.
Examples
Imagine you're designing two differently sized posters that need to have the same proportions to look visually consistent. If the smaller poster is 4 inches by 6 inches, and you want the larger poster to have a perimeter 1.5 times bigger, you can use similar rectangle principles to find the dimensions of the larger poster. This ensures that the posters look like scaled versions of each other, maintaining the same visual appeal regardless of size.
