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.
+128470 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
