Relation Between Side Lengths And Median

Geometry Level 3

In $\triangle{ABC}$ , $D$ is the midpoint of $BC$ . Given that $AB=3$ cm, $AC=5$ cm and $BC=7$ cm, $AD$ can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find the value of $m+n$ .

The answer is 21.

1 solution

Alan Enrique Ontiveros Salazar
Sep 16, 2014

Using Apollonius' Theorem on $\triangle ABC$ we get:

$AD=\dfrac{\sqrt{2(AB^2+AC^2)-BC^2}}{2}$

$AD=\dfrac{\sqrt{2(3^2+5^2)-7^2}}{2}$

$AD=\dfrac{\sqrt{2(34)-49}}{2}$

$AD=\dfrac{\sqrt{68-49}}{2}$

$AD=\dfrac{\sqrt{19}}{2}$

Hence our final answer is $19+2=\boxed{21}$ .

Relation Between Side Lengths And Median

The answer is 21.

1 solution

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