# 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
3 years, 3 months ago

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This was a 2013 secodary level olympiad problem in Khulna

- 3 years, 3 months ago

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

- 3 years, 3 months ago

You qualified that time?

- 3 years, 3 months ago

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

- 3 years, 3 months ago

This will be my 1st time

- 3 years, 3 months ago

$$\dfrac{10\sqrt{6}}{3}$$?

- 3 years, 3 months ago

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

- 3 years, 3 months ago

$$\frac { 20 }{ \sqrt { 3 } }$$ 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 )

- 3 years, 3 months ago

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

- 3 years, 3 months ago

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 )

- 3 years, 3 months ago

why AP+PB=AS+BQ

- 3 years, 3 months ago

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

- 3 years, 3 months ago

oh! I forgot it.....shit.......and thanks......

- 3 years, 3 months ago