Question

IN ABC ,m(<)a=35 and m(<)c=77. wht is the longest side of the triangle?

Answer 1

In ABC, $m(<)a = 35$ and $m(<)c = 77$ . What is the longest side of the triangle?

Solution:

Using Apollonius' theorem we have:

$$a = \frac {2}{3} \sqrt {- m _ {a} ^ {2} + 2 m _ {b} ^ {2} + 2 m _ {c} ^ {2}} = \frac {2}{3} \sqrt {2 m _ {b} ^ {2} + 1 0 6 3 3} (1)$$ $$b = \frac {2}{3} \sqrt {- m _ {b} ^ {2} + 2 m _ {a} ^ {2} + 2 m _ {c} ^ {2}} = \frac {2}{3} \sqrt {- m _ {b} ^ {2} + 1 4 3 0 8} (2)$$ $$c = \frac {2}{3} \sqrt {- m _ {c} ^ {2} + 2 m _ {b} ^ {2} + 2 m _ {a} ^ {2}} = \frac {2}{3} \sqrt {2 m _ {b} ^ {2} - 9 4 0 8} (3)$$

(1), (3) $\rightarrow$ a>c

Median $m(<)c$ is more than twice greater than median $m(<)a$ . So, $m(<)a < m(<)b < m(<)c$ (Otherwise triangle doesn't exist).

Using it we have $a > b$ and $b > c$

So, $a > b > c \rightarrow a$ is the longest side.

Answer: $a$ is the longest side.