3500.31 – Circle Around Three Squares


Tags:Problem Set 3Pre-Calculus MathGeometry-Euclidean

Each side of each square in the figure below has length 1 unit. What's the radius of the smallest circle containing the symmetric figure composed of the three squares as shown?

a. 2\sqrt{2}

b. 1.25\sqrt{1.25}

c. 1.25

d. 517\sqrt{17}/16

e. none of these


Solution

Consult the figure below.

We'll design a circle passing through A, B, C, and D. Its center O will be on the line EF bisecting the symmetry line of the figure. Its its radius will be xx.

Note that xx is the hypotenuse of two right triangles, OEB and OFD. Let OF = tt. Then OE = 1+(1t)=2t1 + (1 - t) = 2 - t. Now x2=t2+12x^2 = t^2 + 1^2 and x2=(2t)2+(1/2)2x^2 = (2 – t)^2 + (1/2)^2

Here we go:

t2+1=44t+t2+¼,4t=3+14,t=1316. \begin{aligned} t^2 + 1 &=& 4 - 4t + t^2 + ¼, \\ 4t &=& 3 + \frac{1}{4}, \\ t &=& \frac{13}{16}. \end{aligned}

So, in ∆OFD,

x2=(1316)2+12=169256+256256=425256=2517256,x=51716. \begin{aligned} x^2 &=& (\frac{13}{16})^2 + 1^2 \\ &=& \frac{169}{256} + \frac{256}{256} \\ &=& \frac{425}{256}= \frac{25\cdot17}{256},\\ x &=& \frac{5√17}{16}. \end{aligned}

We see that xx is approximately 1.29, not so very different from answer choice (c), but (d) is exact. Were you to draw the figure with sides of one inch and make a careful measurement you'd get just about an inch and a quarter for the radius.