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Triangle Radius
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Short
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Secondary curriculum
Problem
Solutions
The circles are tangent to the triangle, so the radii shown below meet the triangle at right angles.
The circles are tangent to each other, so the line drawn connecting their centres is parallel to the base of the triangle.
The angle labelled is $30^\text{o}$ because, by the symmetry of the diagram, it is half of the $60^\text{o}$ angle in the equilateral triangle.
This means that in the diagram, there are two right-angled triangles and a rectangle enclosed by the dotted lines, and $2k+2r=1$.
From the blue triangle, $\tan{30}=\dfrac{r}{k}\Rightarrow k=\dfrac{r}{\tan{30}}=\sqrt{3}r$.
So $2k+2r=1\Rightarrow 2\sqrt{3}r+2r=1\Rightarrow r(2\sqrt{3}+2)=1 \Rightarrow r=\dfrac{1}{2+2\sqrt{3}}$
You can find more short problems, arranged by curriculum topic, in our
short problems collection
.