2007 AMC 10A Problems/Problem 15

Revision as of 19:20, 21 December 2018 by Graydove (talk | contribs) (Solution)

Three circles are inscribed in a rectangle of width w and height h as shown. Two of the circles are congruent and are each tangent to two adjacent sides of the rectangle and to each other. The other circle is larger and is tangent to three sides of the rectangle and to the two smaller circles. What the ratio of h to w? Express your answer as a decimal to the nearest hundredth.

Solution

Draw a square connecting the centers of the four small circles of radius $1$. This square has a diagonal of length $6$, as it includes the diameter of the big circle of radius $2$ and two radii of the small circles of radius $1$. Therefore, the side length of this square is \[\frac{6}{\sqrt{2}} = 3\sqrt{2}.\] The radius of the large square has a side length $2$ units larger than the one found by connecting the midpoints, so its side length is \[2 + 3\sqrt{2}\]. The area of this square is $(2+3\sqrt{2})^2 = 22 + 12\sqrt{2}$ $(B).$

See Also

2007 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 10 Problems and Solutions

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