Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.
May be faster ways to do this, but look at the following:
Let the centers of two of the circles lie at (1,0) and (-1,0)
So....to find the center of the third circle let it be located at (0,y)
And by the Pythagorean Theorem we have
y = sqrt [ GD^2 - HD^2 ]
y= sqrt [ 2^2 - 1^2] = sqrt (3)
So the center of the top circle = ( 0, sqrt (3) ) call this point G
Let point E be the center of the circle.....and it will be located at (0, c )
Now let B be the point tangent to the top circle and the leftmost circle
And its coordinates will be the midpoint of CG = [ -1/2 , sqrt (3)/2 )
Let A be the intersection of the apex of the top circle and the y axis.......and its coordinates are
(0, GA + GH ) = ( 0, 1 + sqrt (3) )
And....by similar triangles
(BF) / FD = HE /HD
(sqrt [(3)/2] ) / (3/ 2) = HE / 1 → HE = sqrt (3)/3
So....the radius of the large circle = [AH - HE] = EA =
[1 + sqrt (3)] - [sqrt (3)/3] =
1 + (2/3) sqrt (3)
Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle.
What is the radius of the large circle?
Express your answer as a common fraction in simplest radical form.
Let Radius of the larger circle = R.
Let Radius of the three circles = r = 1.
Let Area of the triangle ABC = \(A_{ABC}\)
Let Area of the triangle AOB = \( A_{AOB}\)
Let AC = CB = BA = 2r
Let AO = OB = R-r
\(\mathbf{A_{ABC} = \ ?}\)
\(\begin{array}{|lrcll|} \hline (\text{Heron}) & s &=& \frac{2r+2r+2r}{2} \\ & &=& 3r \\ & A_{ABC}^2 &=& s(s-2r)(s-2r)(s-2r) \\ & A_{ABC}^2 &=& s(s-2r)^3 \quad & | \quad s = 3r \\ & A_{ABC}^2 &=& 3r(3r -2r)^3 \\ & A_{ABC}^2 &=& 3rr^3 \\ & A_{ABC}^2 &=& 3r^4 \\ & A_{ABC} &=& r^2 \sqrt{3} \\ \hline \end{array}\)
\(\mathbf{A_{AOB} = \ ?} \)
\(\begin{array}{|lrcll|} \hline & A_{AOB} &=& \frac{A_{ABC}}{3} \\ (1) & A_{AOB} &=& \frac{r^2 \sqrt{3}}{3} \\ \hline \end{array}\)
\(\mathbf{(\text{Heron})\ A_{AOB} = \ ?}\)
\(\begin{array}{|rcll|} \hline (\text{Heron}) & s &=& \frac{2r+(R-r)+(R-r)}{2} \\ & s &=& \frac{2R}{2} \\ & s &=& R \\ & A_{AOB}^2 &=& s(s-2r)[s-(R-r)][s-(R-r)] \quad & | \quad s = R \\ & A_{AOB}^2 &=& R(R-2r)[R-R+r)][R-R+r] \\ & A_{AOB}^2 &=& R(R-2r)r^2 \\ (2) & A_{AOB} &=&r\sqrt{R(R-2r)} \\ \hline \end{array}\)
(1) = (2)
\(\begin{array}{|rcll|} \hline \frac{r^2 \sqrt{3}}{3} &=& r\sqrt{R(R-2r)} \\ \frac{r \sqrt{3}}{3} &=& \sqrt{R(R-2r)} \quad & | \quad \text{square both sides}\\ \frac{r^2}{3} &=& R(R-2r) \\ R^2-2rR -\frac{r^2}{3} &=& 0 \quad & | \quad \cdot 3 \\ 3R^2-6rR - r^2 &=& 0 \\ \\ R &=& \frac{6r \pm \sqrt{36r^2-4\cdot 3\cdot (-r^2) } }{2\cdot 3} \\ R &=& \frac{6r \pm \sqrt{36r^2+12r^2 } }{6} \\ R &=& \frac{6r \pm \sqrt{48r^2 } }{6} \\ R &=& \frac{6r \pm 4r\sqrt{3} }{6} \\ R &=& r \pm \frac23 r \sqrt{3} \quad & | \quad R \gt 0 \\ R &=& r + \frac23 r \sqrt{3} \quad & | \quad r = 1 \\ \mathbf{R} & \mathbf{=} & \mathbf{1 + \frac23 \sqrt{3}} \\ \hline \end{array}\)