2 Blue Moons

Geometry Level 5

4 circles are situated between 2 parallel lines and tangent to each other and to the lines as per diagram. The radii of $C_1$ and $C_2$ are 18 and 6 respectively..

If $x$ is the radius of $C_3$ and $y$ is the radius of $C_4$ , find $x + y$ to the nearest 3 decimal places.

The answer is 17.377.

1 solution

Guiseppi Butel
Mar 1, 2015

Here's my solution:

Could anyone give me more detail on how exactly he solved those Pythagorean equations? The last one is especially tricky for me to find in terms of the radii given.

donkey kong - 5 years, 6 months ago

Referring to my diagram: in ABC: AC =18+6 , AB =18 - 6 ; in ADF: AF = 18 + x , AD = 18 - x ; in CEF: CF = 6 + x , CE = 12 + 18 - x , EF = DF - DE ; I hope this helps.

Guiseppi Butel - 5 years, 6 months ago

How did you get the last y equation?

Roni Jaira - 5 years, 5 months ago

@Roni Jaira – Research "Descartes Theorem"

Guiseppi Butel - 5 years, 5 months ago

@Roni Jaira –

Niranjan Khanderia - 4 years, 11 months ago

Excellent solution!!! Well I didn't notice about AD is actually 18 - x and took a long time in this >.<

AccelNano Lim Loong - 6 years, 3 months ago

I use a different method than yours to get y. I construct a line perpendicular to DF cross the centre of C4 until below the F. You can get the 2 values by using Pythagoras theorem and the sum are same with DF.

AccelNano Lim Loong - 6 years, 3 months ago

Not quite clear about your method. Sorry.

Guiseppi Butel - 6 years, 3 months ago

Niranjan Khanderia - 4 years, 11 months ago

draw tangent from two smaller circles to the touching of two big circles apply pythagoras and it can bi solved easily as tangent are per. to radius

jat in - 6 years, 1 month ago

2 Blue Moons

The answer is 17.377.

1 solution

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