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A median of a triangle is equal in length to the geometric mean of the lengths ofthe sides that include it.  If these two sides are 7 and 10, find the length of the side of the triangle to which the median is drawn.

 Dec 24, 2019
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Geometric mean of  7 and 10  =  sqrt  (  7 * 10)  = sqrt (70)  = median length

 

See the  image below

 

Let AD  = BD   = x

 

By the Law of Cosines   we have these equations

 

[ Note angle BDC  is supplemental to angle  ADC...so  cos BDC  = -cos (CDA)   ]

 

7^2  =  70 + x^2  - 2(√70) (x) cos (ADC)

10^2  = 70 + x^2 - 2(√70)(x) cos (BDC)             so.....         

 

7^2  =  70 + x^2  - 2(√70) (x) cos (ADC)

10^2  = 70 + x^2 - 2(√70)(x) (-cos (ADC))         so......

 

 

7^2  =  70 + x^2  - 2(√70) (x) cos (ADC)

10^2  = 70 + x^2  + 2(√70)(x) cos (ADC)                add these    

     

 

7^2 + 10^2 =  140 +  2x^2      simplify

 

149 =  140 +  2x^2

 

9 = 2x^2     divide both sides  by 2

 

x^2=  9/2

 

x =  3/ √2 =   3√2 / 2

 

So....the length of the remaining side  =  3√2  units

 

cool cool cool

 Dec 24, 2019
edited by CPhill  Dec 24, 2019

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