## Dhaka regional 16 P4

For students of class 9-10 (age 14-16)
mdhasib
Posts: 10
Joined: Fri Jul 22, 2016 11:43 am

### Dhaka regional 16 P4

Untitled.png (44.57 KiB) Viewed 1999 times

Thanic Nur Samin
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

### Re: Dhaka regional 16 P4

Let the reflection of \$X\$ across \$Y\$ be \$Z\$. Now, see that \$\angle PQX=\angle PRZ, \angle PQR=\angle PXY=\angle PZR\$ and finally \$PQ=PR\$. These imply that \$\triangle PQX\cong \triangle PRZ\$. So \$QX=RZ\$, and thus \$QX+RX=RZ+RX=XZ=2XY=24\$
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

Absur Khan Siam
Posts: 65
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

### Re: Dhaka regional 16 P4

How can you write \$XZ = 2XY\$?
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

### Re: Dhaka regional 16 P4

As \$Z\$ is defined as the reflected point of \$X\$ with right to \$Y\$, we get \$ XY=YZ\$. From there we get \$XZ=2XY\$.
Frankly, my dear, I don't give a damn.

Absur Khan Siam
Posts: 65
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

### Re: Dhaka regional 16 P4

Sorry!I missed some questions. How \${\angle}PQX = {\angle}PRZ\$ and\${\angle}PQR = {\angle}PXY\$?
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid

Thanic Nur Samin
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

### Re: Dhaka regional 16 P4

Absur Khan Siam wrote:Sorry!I missed some questions. How \${\angle}PQX = {\angle}PRZ\$ and\${\angle}PQR = {\angle}PXY\$?
Learn the properties of cyclic quads. More than \$50\%\$ of all olympiad geometry problems exploit the usage of cyclic quad. https://www.expii.com/t/when-is-a-quadr ... on&id=1795
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.