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A Problem need to solve.....

I need to know how to solve this problem. Can anyone help me?

In trapezium ABCD, ADIIBC, AD<BC,unparallel sides are equal.A circle with centre O is inscribed in the trapezium. OAD is equilateral. Find the radius of the circle if the area of the trapezium is \(\frac{800} {\sqrt{3}}\)

Note by Partho Kunda
2 years, 9 months ago

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This was a 2013 secodary level olympiad problem in Khulna

Rifath Rahman - 2 years, 9 months ago

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You're right.....Good memory of course.......

Partho Kunda - 2 years, 9 months ago

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You qualified that time?

Rifath Rahman - 2 years, 9 months ago

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@Rifath Rahman Yeah! 1st Runners up.....I qualified in the regional for 4 times.....

Partho Kunda - 2 years, 9 months ago

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@Partho Kunda This will be my 1st time

Rifath Rahman - 2 years, 9 months ago

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\(\dfrac{10\sqrt{6}}{3}\)?

Krishna Sharma - 2 years, 9 months ago

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I really don't know. Do u know how to solve it?

Partho Kunda - 2 years, 9 months ago

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\(\frac { 20 }{ \sqrt { 3 } }\) should be the answer. (by the way, I wrote up a solution and God knows where I made the mistake in LATEXing. Everything started looking like Hebrew :P )

Homo Sapiens - 2 years, 9 months ago

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I really don't know. Do u know how to solve it?

Partho Kunda - 2 years, 9 months ago

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drop perpendiculars from A & D to BC and call the feet M and N respectively. Let r be the radius and AD = a Say, the circle touches AB . BC , CD , DA at P, Q, R, S respectively, Here AM = 2r and AS = MQ = a/2

Now, OS is the height of equilateral triangle AOD , so OS =r = (root3)a/2 and, AB = AP + PB = AS + BQ = AS + BM +MQ = a + BM [ as AS = MQ = a/2 ] now use Pythagoras in ABM and you'll get a relation between BM and r

so, area of ABCD = ABM + DNC + AMND = 2ABM + AMND [since AMB and DNC are congruent ] Solve the equation and you'll get the answer

(Hope you'll understand..... I messed up again trying to LATEX :P )

Homo Sapiens - 2 years, 9 months ago

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@Homo Sapiens why AP+PB=AS+BQ

Partho Kunda - 2 years, 9 months ago

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@Partho Kunda If two tangents are drawn to a circle from an external point, the distances from that point to the points of contact are equal.

Homo Sapiens - 2 years, 9 months ago

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@Homo Sapiens oh! I forgot it.....shit.......and thanks......

Partho Kunda - 2 years, 9 months ago

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