Fundamental Drill#6

Geometry Level 3

As shown, $PQRS$ is a cyclic quadrilateral and the segments $PQ,QR,RS$ have equal length.

Given that the radius of the circle is $3$ and $\angle PSR=20^\circ$ . Find the length of segment $PS$ .

You can try more of my fundamental problems here .

2 solutions

Donglin Loo
Jun 23, 2018

$\because \overline{PQ}=\overline{QR}=\overline{RS}$

$\therefore \angle POQ=\angle QOR=\angle ROS$

Let $\angle POQ=\angle QOR=\angle ROS=x^\circ$

$\angle PSR=\cfrac{1}{2}\cdot \angle POR$

$20^\circ=\cfrac{1}{2}\cdot \angle POR$

$\angle POR=40^\circ$

$\angle POR=2x^\circ$

$2x^\circ=40^\circ$

$x=20$

$\angle POS=3x^\circ=3\cdot 20^\circ=60^\circ$

$PO,OS$ are radii of the circle.

$\overline{PO}=\overline{OS}$

$\because \angle POS=60^\circ$

$\therefore$ $\Delta POS$ is an equilateral triangle.

$\overline{PS}=\overline{PO}=\overline{OS}=3$

Can you explain the PSR=POR/2 ?

Long Plays - 2 years, 11 months ago

@Long Plays Everything That's called the central angle theorem. You can go look up for it in the internet

donglin loo - 2 years, 11 months ago

X X
Jun 23, 2018

$\angle POR=20^\circ \times 2=40^\circ,\angle POS=40^\circ\times \frac32=60^\circ$ ,so $PS=PO=3$