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I am working on this question, and the solution says something weird.

Here is the question enter image description here:

The solution starts by saying

enter image description here

The part I don't get is where they say 'PQR is $9 = 3^2$ to $1$, so the ratio is $3$ to $1$'. How can you square root each side of a triangle and keep it proportional to the other? If you multiplied each side by a number like $4$, the proportionality would stay the same, but if you squared or square rooted, wouldn't the triangles be no longer proportional?

Thanks for the help!

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  • $\begingroup$ Read carefully: proportion $9:1$ is referred to the areas of triangles. And areas are indeed proportional to the squares of lengths. $\endgroup$ – Aretino Feb 10 at 15:12
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You are correct, the ratio of the lengths of a single shape is the same for any two similar shapes, so one cannot square or square root the lengths and have this ratio the same. This question is different - it is talking about the area ratio between similar shapes. If we square root the ratio of the areas between similar shapes we get the corresponding ratio of the lengths between the shapes.

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