Question

the sides of a triangle are 5 cm; 7 cm and 10 cm. find the length of the median to the longest side.

Answer 1

The length of the median to the longest side will be $2\sqrt{3}$ cm.

Explanation

Suppose, the sides of the triangle are  $a, b$ and $c$

So, $a=5 cm, b= 7cm$ and $c= 10 cm$

Here the length of the longest side is 'c'

Formula for finding the length of the median to side 'c' is........

$m_{c}=\frac{1}{2}\sqrt{2(a^2+b^2)-c^2}$

So......

$m_{c}= \frac{1}{2}\sqrt{2(5^2+7^2)-10^2}\\ \\ m_{c}=\frac{1}{2}\sqrt{2(25+49)-100}\\ \\ m_{c}= \frac{1}{2}\sqrt{148-100} =\frac{1}{2}\sqrt{48}=\frac{1}{2}*4\sqrt{3}=2\sqrt{3}$

Thus, the length of the median to the longest side will be $2\sqrt{3}$ cm.

Answer 2

Using heron's formula in a triangle ABC, D is the point of median from B then (AB)^2 + (BC)^2 = 2*(BD^2 + DC^2 )

Step-by-step explanation:here using above formula

25+49 = 2*(x^2 +25)

Solving x= 2(root 3)