7150
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2011-02-01T00:00:01
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<mdoxml version="1.0"><p style="text-align: left;">The diagram shows two identical large circles and two identical smaller circles whose centres are at the corners of a square. The two large circles are touching, and they each touch the two smaller circles. The radius of the small circles is $1cm$. What is the radius of the large circles in centimetres?</p>
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If you liked this problem, <a href="http://nrich.maths.org/372&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br>
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Let $r$ be the radius of each of the larger circles.<br></br>
The sides of the square are equal to $r+1$, the sum of the two
radii.<br></br>
The diagonal of the square is $2r$.<br></br>
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By Pythagoras, $$(r+1)^2+(r+1)^2 =
(2r)^2$$Simplifying gives: $$
2(r+1)^2 = 4 r^2$$ i.e.
$$(r+1)^2 =
2r^2$$<br></br>
so$$r+1 = \sqrt{2}r$$<br></br>
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[$-\sqrt{2}r$ is not possible since $r+1> 0$].<br></br>
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Therefore $(\sqrt{2} - 1)r = 1$. <br></br>
Hence $r= \frac {1}{\sqrt{2}-1} = \sqrt{2} + 1$.<br></br>
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Weekly Problem 3 - 2011
Weekly Problem 3 - 2011
2D Geometry, Shape and Space
Pythagoras' theorem
Algebra
Quadratic equations
Using, Applying and Reasoning about Mathematics
Visualising
Secondary Mapping Document
Pythagoras’s Theorem