Bader A.

asked • 09/28/18

Pre calculus question !!!

In a triangle ΔABC , it is known that AB =10 , AC = 9 , cos 3\8 
∠BAC = . Points P, Q on sides AB , AC such that the area of ΔAPQ is half of the area of ΔABC . For all points P, Q satisfying the above, the smallest possible value of PQ is  :
 
(a) 7        (b) 15\2       (c) 8   (d) 17\2    (e) 19\2  
 

Victoria V.

I tried several methods including setting up an excel spreadsheet and trying various values of AP and AQ, constrained by the fact that the area of APQ had to be half of the area of ABC.  The spread sheet told me the correct answer was 7.50000086, and if I had narrowed it down further, or not rounded, it would probably be exactly 7.5 after many more iterations.  So I would choose (b)  15/2
 
The only other way I could figure out how to solve it was by using Geometery.
Learn in Geom that the ratio of Areas is the square of the ratio of the lengths of the sides.
So I calculated the length of BC =sqrt(227/2) = approx 10.65364
 
A small = 1
--------------
A big     = 2
 
So
Side small = √1
-------------------
Side big     = √2
 
So set up the ratio:
 
 
small side = 1       PQ
---------------- = -------
big side   = √2         10.65364
 
Solving I get PQ = 10.65364/√2   = approx 7.533
 
So again, this is not exactly 15/2, but it is the closest to 15/2, and 15/2 is actually smaller, so it would be the smallest possible value.
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09/28/18

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Victoria V. answered • 09/28/18

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what is meant by    
 
cos 3\8 ∠BAC = .
 
 
?????

Have you gotten either of my comments? As soon as I hit "SUBMIT" they disappear, and I really do not want to type it again, if you have already gotten it twice,... but here it is, again maybe. I tried many methods and finally put together an Excel spreadsheet with various values for the sides AP and AQ, constrained by the fact the the area of the smaller is 1/2 the area of the larger, which resulted in that (AP)(AQ)=45. It said that the smallest PQ is 7.50000086 So I would choose the answer 15/2

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Bader A.

Sorry i meant Cos ∠BAC = 3/8 
Verry sorrryyy !!!
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09/28/18

Victoria V.

I have tried this many ways, I finally set up an excel spreadsheet and tried many numbers.  It said the smallest PQ could be was 7.50000086, so I would say the answer is 15/2
 
The way I finally worked it out was with Geometry.  If the ratio of Areas is x/y, then the ratio of the sides is √x⁄√y
BC, using the law of cosines, has a length of sqrt(277/2) = approx 10.65364
 
A big = 2            BC (big) = √2       10.65364
-----------   so   ------------------ =  ------------
Asmall=1            PQ (small) = 1          ???
 
Solving this proportion, I get   10.65364/√2 = approx 7.533, so again I would choose 15/2    
 
There is probably a better way to solve it, but this was what I could come up with.
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09/28/18

Bader A.

Yess yess ,, thank you so much ma'am  I got all your comments and replies ,, thank you sooo sooo much 
 
When i first answered this question , i thought it was 7 , but i think your answer is more accurate, thank you again !!
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09/29/18

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