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bob bundy
2013-04-20 01:01:21

Just to finish it off:

QED.

Bob

anonimnystefy
2013-04-20 00:10:00

Yes, sorry. I edited it.

bobbym
2013-04-20 00:03:00

Don't you mean 108?

anonimnystefy
2013-04-19 23:59:43

Okay. If you leave the surds as is without simplification you will also get the exact answer of 108.

bob bundy
2013-04-19 22:03:02

hi Stefy,

I just did 1/2 product of the diagonals.  Sketchpad does have an area function, but I've never used it.

Bob

anonimnystefy
2013-04-19 18:41:32

Ah, I get it. Great! Do you use SketchPad to get the area?

bob bundy
2013-04-19 17:29:16

hi Stefy,

There are three equations ( two from cosine rule and the straight line equation) and there are three unknowns (R,r and alpha).  So it should be possible to eliminate two unknowns to find the third.  Both the cosine rule equations have the same term in cos alpha but in one it is negative.  So adding the equations together (LHS1 + LHS2 = RHS1 + RHS2) gets alpha out straight away.  But I eliminated  R first before the adding so I had a quadratic in r straight away.

Thanks for giving me the start.  I had stared at that diagram for ages without thinking of that.  Too distracted trying to use angle properties.

I think the expression for r and R could be left in surd form and then the area obtained without any recourse to decimals.  Might try it later.  That way you get to show the answer is exactly 180.

Bob

anonimnystefy
2013-04-19 08:03:18

Hm, how did you get the second to last row?

bob bundy
2013-04-19 06:43:01

hi Stefy,

Your idea and then I followed a different path:

Triangle DBA

Triangle ABC

Using R = 12 + r this becomes

Bob

anonimnystefy
2013-04-19 06:09:04

Okay. Post when you get the equations.

bob bundy
2013-04-19 06:07:41

Thanks.  Working on that now.

B

anonimnystefy
2013-04-19 06:01:20

ABD with the angle alpha and ABC with angle 180-alpha.

bob bundy
2013-04-19 05:57:31

hi Stefy,

Sounds brilliant but too many steps missing for my little brain.

Which triangle(s) are you using for the cosine rule?

Bob

anonimnystefy
2013-04-19 05:25:00

Well, I just noticed that the point where the circles touch is on the same line as A and B (because they are the centers of the two circles), and got from that that R=AB+r, that is, AC=AB+BD. Now we can use the Cosine Rule to get

,
where alpha is the acute angle of the rhombus. We can get from that equation the cosine of the angle and thus also its sine. We then use the formula for calculating the area of parallelogram using its sides and internal angle.

bobbym
2013-04-19 05:11:21

It might help Bob to know what the correct answer is. I think once he gets the sides to 12 he will get it.