6265 /www/nrich/html/content/id/6265/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <p>$ABCD$ is a square. $P$ and $Q$ are points outside the square such that triangles $ABP$ and $BCQ$ are both equilateral. How big is angle $PQB$?</p> <p> </p> <p>If you liked this problem, <a href="http://nrich.maths.org/2845">here is an NRICH task</a> which challenges you to use similar mathematical ideas.</p> <p> </p> <p> </p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><mdo:image alt="solution" height="200" src="solution.png" width="200"></mdo:image><br></br> <br></br> Since $ABQ$ and $BCQ$ are equilateral, the angles $ABP$ and $CBQ$ are both $60^\circ$. So $$\angle{PBQ} = 360^\circ-90^\circ-60^\circ-60^\circ=150^\circ$$<br></br> PBQ is isosceles, so the angles $BPQ$ and $PQB$ are equal. So<br></br> $$2 \times \angle{PQB} = 180^\circ- 150^\circ = 30^\circ$$ So $$\angle{PQB} = 15^\circ$$</p></mdoxml> 2 3 0 0 1 0 0 ABCD is a square. P and Q are points outside the square such that triangles ABP and BCQ are both equilateral. How big is angle PQB? 2D Geometry, Shape and Space Angles 2D Geometry, Shape and Space Equilateral triangles