25. The diagram shows three touching semicircles with radius 1 inside an equilateral triangle, which each semicircle also touches. The diameter of each semicircle lies along a side of the triangle.
What is the length of each side of the equilateral triangle?
A 3
B $\frac{13}{4}$
C $\frac{3}{2} \sqrt{3}$
D $2 \sqrt{3}$
E 4
25. D In the diagram, $P, Q$ and $R$ are the vertices of the triangle; $S$ and $T$ are both centres of a semicircle; $U$ is a point where two semicircles touch and $V$ and $W$ are points where a semicircle touches the triangle. So angles $P V S$ and $T W S$ are right angles, since a tangent to a circle is perpendicular to the radius of that circle at the point of contact. Also, the line joining the centres of two touching semicircles passes through their point of contact, so $S U T$ is a straight line and has length
2. Note that $\angle V P S=60^{\circ}$, and so $\angle P S V=30^{\circ}$.
Note also that $S T=2, T W=1$ and $\angle T W S$ is a right angle.
Therefore $\sin \angle W S T=\frac{1}{2}$ and so $\angle W S T=30^{\circ}$. However $\angle T Q S=60^{\circ}$ and so $\angle S T Q$ is a right angle.
From triangle $S P V$, we see that $\frac{S V}{P S}=\frac{1}{P S}=\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ and so $P S=\frac{2}{\sqrt{3}}$. Similarly, from triangle $S Q T$, we see that $S Q=\frac{4}{\sqrt{3}}$. Therefore $P Q=\frac{6}{\sqrt{3}}=\frac{6 \sqrt{3}}{3}=2 \sqrt{3}$.