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In triangle $ABC$, $AB = 7$, $AC = 17$, and the length of median $AM$ is $12$. Find the area of triangle $ABC$.

 Jan 23, 2025
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                  A

 

         7           17        

                12

B              M                 C

 

BM = CM

 

Law of Cosines

 

Note : cos CMA =  - cos BMA

We have these two equations

7^2 = 12^2 + BM^2  - 2(12 * BM) cos (BMA)

17^2  - 12^2  + BM^2 - 2(12 *BM) cos (CMA)     

 

7^2 = 12^2 + BM^2  - 2(12 * BM) cos (BMA)

17^2  = 12^2  + BM^2 + 2(12 *BM) cos (BMA)    add these

 

338 = 288 + 2BM^2

 

50 = 2BM^2

 

25 = BM^2

 

BM = 5 = CM

BC = 10

 

However, this triangle is  impossible because  BC + AB  must be >  AC

 

cool cool cool

 Jan 24, 2025

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