square paper folding (May Olympiad 2013 L1)

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High School Math Grades 9-12, Ages 13-18

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#1 h May 11, 2019, 3:02 AM • 1 Y

Y by Adventure10

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$ . By folding the paper along the $AP$ line, point $B$ determines the point $Q$ , as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$ . Calculate the perimeter of the triangle $PCR$

https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png

This post has been edited 3 times. Last edited by parmenides51, Jun 3, 2024, 5:14 PM

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#2 h May 11, 2019, 6:53 PM • 2 Y

Y by Adventure10, Mango247

$A(0,0),B(0,10),C(10,10),D(10,0)$ .
Let $P(\alpha,10)$ .

Then $Q(\frac{200\alpha}{100+\alpha^{2}},\frac{1000-10\alpha^{2}}{100+\alpha^{2}})$
and $R(10,\frac{100-10\alpha}{10+\alpha})$ .

Distances $PC=10-\alpha, CR=\frac{20\alpha}{10+\alpha},PR=\frac{100+\alpha^{2}}{10+\alpha}$ .
Perimeter $=20$ .

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#3 h Jun 3, 2024, 4:44 PM

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