+1859 In the diagram, ABCD is a square. Find PR.
L, M, N, O, are midpoints of sides.
Side AB = 12
0 \(PR \) 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
\(\Large\frac{PR}{AC}=\frac{x}{DA}\\ \boxed{PR=\frac{12\sqrt2}{\sqrt5}}\)
