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In the diagram, ABCD is a square.  Find PR.

 

 

L, M, N, O, are midpoints of sides.
Side AB = 12

 Feb 16, 2024
 #1
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\(PR \) is the diagonal of the square \(PQRS\). So we can find the sidelength of this square and it will be in the same ratio as the diagonal of the square \(ABCD\).

\(\triangle DOS\sim\triangle DAP\) Therefore, 

\(\frac{DO}{DA}=\frac{DS}{DP}\\ \Rightarrow DP=2DS\)
\(DP-SP=DS\\\Rightarrow SP=DS\)
\(SP\) is the side of the triangle and lets name it \(x\). Due to symmetry, \(AP=x\) so, 

\(AP^2+(DP)^2=DA^2\\ 5x^2=144\\ x=\frac{12}{\sqrt5}\)
 

thus the ratio of diagonals = ratio of sides as all sqaure are similar

\(\Large\frac{PR}{AC}=\frac{x}{DA}\\ \boxed{PR=\frac{12\sqrt2}{\sqrt5}}\)

 Feb 16, 2024
 #2
avatar+1622 
+2

Nice! I agree with your answer: personally, I prefer rationalizing the denominator: \({12\sqrt{2}\over\sqrt{5}}={12\sqrt{2}\over\sqrt{5}}*{\sqrt{5}\over\sqrt{5}}={12\sqrt{10}\over5}\)smiley

proyaop  Feb 16, 2024

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