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Points S and T are on side CD of rectangle ABCD such that AS and AT trisect angle DAB . If CT=3 and DS=6, then what is the area of ABCD?

Mellie  Nov 14, 2015

Best Answer 

 #2
avatar+78618 
+15

Nice, Melody.....

 

ST can also be found, thusly :

 

AD = 6/tan(30) = 6sqrt(3)

 

And, using similar triangles, we have

 

DS / AD  ≈ AD / [DS + ST]

 

DS^2 + DS*ST   = AD^2

 

ST  = [ AD^2 - DS^2] /  DS

 

ST = [ 108 - 36] / 6  =  [ 72 / 6] = 12

 

And the area is :

 

[DS + ST + TC] * AD =

 

[6 + 12 + 3] * 6/tan (30) =  126/tan(30) =  126*sqrt(3)  units^2

 

 

 

cool cool cool

CPhill  Nov 16, 2015
edited by CPhill  Nov 16, 2015
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3+0 Answers

 #1
avatar+91001 
+15

Hi Mellie,

I love answering your questions. I would of answered this one earlier but it got lost. I thought I saw it but then I could not find it.     angry

 

Points S and T are on side CD of rectangle ABCD such that AS and AT trisect angle DAB . If CT=3 and DS=6, then what is the area of ABCD?

 

Here is the diagram. I answered it just  from a rough sketch but i wanted to check it was correct so I draw it to scale. :)

 

 

 

 DS is given as 6 units and TC is given as 3 units, ST is not given.

 

The LaTex function is not working so I will have to do it by hand.

 

tan 30 = DS/H = 6/H

1/sqrt3 = 6/H

H=6sqrt3

 

tan60 = DT/H

sqrt3 = (6+ST)/6sqrt3

6*3=6+ST

18=6+ST

12=ST

 

so

DC=6+12+3 = 21

Area = AD * DC

Area = 6sqrt3 * 21

Area = 126 sqrt3      units squared.

 

That was a great question - thanks Mellie.

Melody  Nov 15, 2015
 #2
avatar+78618 
+15
Best Answer

Nice, Melody.....

 

ST can also be found, thusly :

 

AD = 6/tan(30) = 6sqrt(3)

 

And, using similar triangles, we have

 

DS / AD  ≈ AD / [DS + ST]

 

DS^2 + DS*ST   = AD^2

 

ST  = [ AD^2 - DS^2] /  DS

 

ST = [ 108 - 36] / 6  =  [ 72 / 6] = 12

 

And the area is :

 

[DS + ST + TC] * AD =

 

[6 + 12 + 3] * 6/tan (30) =  126/tan(30) =  126*sqrt(3)  units^2

 

 

 

cool cool cool

CPhill  Nov 16, 2015
edited by CPhill  Nov 16, 2015
 #3
avatar+91001 
+5

Thanks CPhill, that is a good answer also.   laugh

Melody  Nov 17, 2015

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