Question

rectangles abcd and efgh are similar. the length of each side of efgh is 7 times the length of the corresponding side of abcd. the area of abcd is 84 square units. what is the area, in square units, of efgh?
a. 12
b. 49
c. 588
d. 4,116

Explanation:

Step1: Recall the property of similar figures.

For similar figures, the ratio of their areas is the square of the ratio of their corresponding side lengths. Let the ratio of side lengths of EFGH to ABCD be \( k = 7 \).

Step2: Calculate the ratio of areas.

The ratio of the area of EFGH to ABCD is \( k^2 = 7^2=49 \).

Step3: Find the area of EFGH.

Let \( A_{ABCD} = 84 \) and \( A_{EFGH} \) be the area of EFGH. Then \( \frac{A_{EFGH}}{A_{ABCD}}=49 \), so \( A_{EFGH}=49\times A_{ABCD} \). Substituting \( A_{ABCD} = 84 \), we get \( A_{EFGH}=49\times84 = 4116 \)? Wait, no, wait. Wait, the problem says "the length of each side of EFGH is 7 times the length of the corresponding side of ABCD". Wait, maybe I misread. Wait, similar rectangles: if the scale factor (ratio of sides) is \( k \), then area ratio is \( k^2 \). Wait, but if EFGH's side is 7 times ABCD's, then scale factor \( k = 7 \), so area of EFGH is \( k^2\times \) area of ABCD? Wait, no, wait: if ABCD has side length \( s \), area \( A = s\times l \) (for rectangle). EFGH has side length \( 7s \) and \( 7l \) (since similar, so both sides scaled by 7). Then area of EFGH is \( 7s\times7l = 49sl = 49\times A_{ABCD} \). But \( A_{ABCD} = 84 \), so \( A_{EFGH}=49\times84 = 4116 \)? But wait, the options are a.12, b.49, c.588, d.4116. Wait, 49×84: 50×84=4200, minus 1×84=84, so 4200 - 84 = 4116. But wait, maybe I made a mistake. Wait, no, wait: maybe the scale factor is EFGH to ABCD is 7, so area ratio is 7²=49, so area of EFGH is 49×84? Wait, but 84×49: 80×49=3920, 4×49=196, 3920+196=4116. So the answer should be d. But wait, let me check again.

Wait, the problem says "Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 7 times the length of the corresponding side of ABCD. The area of ABCD is 84 square units. What is the area of EFGH?"

So, for similar rectangles, if the ratio of corresponding sides (scale factor) is \( k \), then the ratio of their areas is \( k^2 \). Here, \( k = 7 \) (since EFGH's side is 7 times ABCD's). So area ratio is \( 7^2 = 49 \). Therefore, area of EFGH = 49 × area of ABCD = 49 × 84 = 4116. So the answer is d.

Answer:

d. 4,116