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 $\frac{800} {\sqrt{3}}$

#Geometry

Easy Math Editor

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Comments

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

Krishna Sharma - 6 years, 5 months ago

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

Partho Kunda - 6 years, 5 months ago

This was a 2013 secodary level olympiad problem in Khulna

Rifath Rahman - 6 years, 5 months ago

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

Partho Kunda - 6 years, 5 months ago

You qualified that time?

Rifath Rahman - 6 years, 5 months ago

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

Partho Kunda - 6 years, 5 months ago

@Partho Kunda – This will be my 1st time

Rifath Rahman - 6 years, 5 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 )

Homo Sapiens - 6 years, 5 months ago

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

Partho Kunda - 6 years, 5 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 )

Homo Sapiens - 6 years, 5 months ago

@Homo Sapiens – why AP+PB=AS+BQ

Partho Kunda - 6 years, 5 months ago

@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 - 6 years, 5 months ago

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

Partho Kunda - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$