# Splitting a triangle to make two equal halves, find the length of the new line

Could someone please explain to me how I would find this out?

I have a triangle and I need to find the length of the line that would split it down the middle so that the areas were even.

A = 105 degrees

B = 42 degrees

C = 33 degrees

AB = 3.2

AC = 3.9

BC = 5.7

The line needs to be draw across the triangle splitting angle A (the 105 degrees angle).

The total area of the triangle is $6.075km^2$ , so the two halves have to be equal in area.

Can anyone help me out here? Thank you!

• A truncation error occurs in your data set. Assuming that all the angles, plus AB = 3.2 and BC = 5.7 are all correct, then AC = .. by cosine law .. = 3.9(522). There are other combinations too. Should the correct combination be used, it is possible to get an answer matching that from the book. – Mick Aug 4 '14 at 4:04

The line has to be drawn just through the midpoint of $BC$.
To find the length of a median you can use the formula: $$m_a^2 = \frac{2b^2+2c^2-a^2}{4},$$ giving in your case: $$m_a = \frac{1}{2}\sqrt{2\cdot 3.9^2+2\cdot 3.2^2-5.7^2}=2.145\ldots$$
• @StevenStadnicki is clearly right, the shortest height has length $\frac{2\cdot6.075}{5.7}=2.13\ldots$. – Jack D'Aurizio Aug 4 '14 at 3:36