Question

question
rectangles s and t are similar. if the area of rectangle s is 85, what is the area of rectangle t?
rectangle s
area = 85
17
rectangle t
7.65
answer attempt 1 out of 2
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Explanation:

Step1: Find the scale factor

The corresponding sides of similar rectangles S and T are 17 and 7.65. The scale factor \( k \) is the ratio of the side of T to the side of S, so \( k=\frac{7.65}{17} = 0.45\).

Step2: Relate areas of similar figures

For similar figures, the ratio of their areas is the square of the scale factor. Let \( A_S \) be the area of S and \( A_T \) be the area of T. So \( \frac{A_T}{A_S}=k^2 \).

Step3: Calculate the area of T

We know \( A_S = 85 \) and \( k = 0.45 \), so \( k^2=(0.45)^2 = 0.2025 \). Then \( A_T=A_S\times k^2=85\times0.2025 = 17.2125 \)? Wait, no, wait. Wait, maybe the side given for S is length, let's check the length of S's other side. Wait, area of S is 85, one side is 17, so the other side of S is \( \frac{85}{17}=5 \). Then the sides of S are 17 and 5. The corresponding side of T is 7.65, so the scale factor is \( \frac{7.65}{17}=0.45 \), so the other side of T is \( 5\times0.45 = 2.25 \). Then area of T is \( 7.65\times2.25 = 17.2125 \)? Wait, no, maybe I made a mistake. Wait, 17 times 5 is 85, correct. 7.65 is 170.45, so the other side is 50.45=2.25. Then 7.652.25: 72.25=15.75, 0.652.25=1.4625, total 15.75+1.4625=17.2125. Wait, but maybe the scale factor is squared for area. Wait, area ratio is (scale factor)^2. Scale factor is 0.45, so area ratio is 0.2025, 850.2025=17.2125. Alternatively, maybe the sides are 17 (length) and 5 (width) for S, and for T, length is 7.65, width is 5(7.65/17)=50.45=2.25, so area is 7.652.25=17.2125. So the area of T is 17.2125? Wait, but maybe the problem is that the side given for T is 7.65, which is the length, and the other side is scaled. Wait, maybe I miscalculated. Wait, 17 divided by 7.65 is approximately 2.222..., wait no, 7.65 is less than 17, so scale factor is 7.65/17=0.45. So area ratio is (0.45)^2=0.2025, 850.2025=17.2125. So the area of T is 17.2125.

Answer:

17.2125