Two exterior triangles
Weekly Problem 35 - 2009
Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?
Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?
Age
11 to 14
Challenge level
Problem
A square is labelled clockwise $ABCD$. $P$ and $Q$ are points outside the square such that triangles $ABP$ and $BCQ$ are both equilateral.
How big is angle $PQB$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Image
Since $ABQ$ and $BCQ$ are equilateral, the angles $ABP$ and $CBQ$ are both $60^\circ$. So $$\angle{PBQ} = 360^\circ-90^\circ-60^\circ-60^\circ=150^\circ$$
PBQ is isosceles, so the angles $BPQ$ and $PQB$ are equal. So
$$2 \times \angle{PQB} = 180^\circ- 150^\circ = 30^\circ$$ So $$\angle{PQB} = 15^\circ$$