How to calculate the area of the rectangle | 3 Circles insides a rectangle | Tricky Math Question

[@GoogleAccount-pi9ct]

Pretty common trick, but pretty cool. Can I suggest covering AIME II 2023 Problem 13 and USAMO 2010 Problem 1?

[@LearnMathwithZain]

OK, I will try to look for it. Thank you for your comment. 🌹

[@glizio5667]

I chose to take the altitude of the equilateral triangle created after connecting the 3 radii. 30-60-90 so it’s rad3 and then from there it’s just 2+ rad3 as the height and 4 length. 4(2 + rad3) = 8 + 4rad3

[@LearnMathwithZain]

Great

[@SOURAVKUMAR-tt7bx]

But sir i have a doubt how can we say that that the vertical line will be tangent to the bottom circle i know it seems pretty obvious by looking st the symmetry but can you still tell me the mathmatical reason that why the vertical line is tangent to the bottom circle

[@LearnMathwithZain]

The radii of all the circles are 2. If we draw a line, parallel to the the side of the rectangle, passing from the center of the circle on the top will be tangent to the circle at the bottom, because it has also radius 2. I hope it is clear now. Focus on the radius of the top and bottom circle, it will be clear to you.