The Gemstone

Geometry Level 3

If all circles with centers A , B , C , D , E A, B, C, D, E and F F have the same radius of 7, what is the area of the blue circle squeezed into the yellow shaded region given that circles E E and F F passes through the centers of the other circles and that ABCD forms a rectangle (when joining up those points with lines).

Give your answer to 2 decimal places.

Easier problem .


The answer is 16.04783885.

Samy Adams by Samy Adams , 23, United Kingdom

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3 solutions

Samy Adams
Jun 9, 2016

Here is what I did to find the radius of the small blue circle by doing (length of larger hypotenuse) - 7(2) = 2r and then I simply used that radius to find the area using area of a circle formula.

Note: The 60 degrees arises from the fact that the equal radii create an equilateral triangle.

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Bear in mind this quiz question would not pass mustard on an actual test. It was not stated that the distance between AC and BD circles were such that they intersected precisely with EF. This was assumed from diagram but not stated. EF would move apart or closer together vertically if the other circles moved horizontality.

Emmery Neuperger - 4 years, 10 months ago

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Sorry if I haven't included anything essential. I did mention that circles E and F passes through the centers of the other circles and I've mentioned that all circles have equal radii. Therefore, shouldn't this mean that circles A,B,C & D intersect with the centers of E and F? I thought this because for example, circle center A is 7 units distance from circle center E (for circle E to have its circumference intersect circle center A) and circle A has the same radius as circle E (of 7 units) so this means that circle center E must lie on the circumference of circle A as circle A has a radius of 7 and center E is 7 units away from center A. Therefore, this means that they intersect.

Also, E and F cannot move apart or closer together vertically as the two circles intersect one anothers centers hence, "E and F passes through the centers of the other circles". The "other circles" include one another too.

Samy Adams - 4 years, 10 months ago
Ahmad Saad
Jun 10, 2016

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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