angle wanted inside a rectangle, related to 20^o (2000 Argentina OMA L2 p2)

Art of Problem Solving

AoPS Online

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

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

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

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

angle wanted inside a rectangle, related to 20^o (2000 Argentina OMA L2 p2)

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30650 posts

#1 h Jun 10, 2020, 6:21 AM

Y by

Let $ABCD$ be a rectangle, $M$ the midpoint of $DA$ , $N$ the midpoint of $BC$ . Let $P$ be the point on the extension of the side $CD$ (closer to $D$ than to $C$ ) such that $\angle CPN = 20^o$ . Let $Q$ be the point of intersection of the line $PM$ with the diagonal $AC$ . Calculate the measure of the angle $\angle PNQ$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

9003 posts

#2 h Jun 10, 2020, 3:29 PM

Y by

Let $A(0,0),B(b,0),C(b,c),D(0,c)$ , then $M(0,\frac{c}{2}),N(b,\frac{c}{2})$ .
Choose $P(-\lambda,c)$ , then $\tan 20^{\circ}=\frac{c}{2(b+\lambda)}$ , from which $\lambda=\frac{c}{2\tan 20^{\circ}}-b$ .

Equation of $AC\ :\ y=\frac{c}{b}x$ , equation of $PM\ :\ y=\frac{c}{2}-\frac{cx}{2\lambda}$ .
Both lines cut in the point $Q(\frac{b\lambda}{2\lambda+b},\frac{c\lambda}{2\lambda+b})$ .

Slope of the line $PN\ :\ m_{PN}=-\frac{\frac{c}{2}}{\lambda+b}$ , slope of the line $QN\ :\ m_{QN}=\frac{\frac{c}{2}}{\lambda+b}$ .

$\tan \widehat{PNQ}=\frac{\frac{\frac{c}{2}}{\lambda+b}+\frac{\frac{c}{2}}{\lambda+b}}{1-\frac{\frac{c}{2}}{\lambda+b} \cdot \frac{\frac{c}{2}}{\lambda+b}}$ ,
$\tan \widehat{PNQ}=\frac{4c(\lambda+b)}{4(\lambda+b)^{2}-c^{2}}$ ,
$\tan \widehat{PNQ}=\frac{2\tan 20^{\circ} }{1-\tan^{2}20^{\circ}}=\tan 40^{\circ}$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

4388 posts

#3 h Jun 15, 2020, 7:46 PM • 1 Y

Y by Mango247

The problem can be generalized.

Best regards,
sunken rock

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a