Thread: Calculating the base of a triangle, with median and legs

1. Calculating the base of a triangle, with median and legs

Hi Forum,

I searched in some places and I couldn't find a good way to do this. The way the solution is presented is not very clear.

In the triangle ABC, AB=4 and AC=8. If M is the mid point of BC and AM=3 what is the length of BC?

Code:

A
4                    8
3
x           y
B          M           C
Now, if M is the midpoint between B and C then x and y are equal.
The solution considers them as two different numbers.
AM is the median, right?
How should I proceed now?

Thanks!

2. Originally Posted by Zellator
Hi Forum,

I searched in some places and I couldn't find a good way to do this. The way the solution is presented is not very clear.

In the triangle ABC, AB=4 and AC=8. If M is the mid point of BC and AM=3 what is the length of BC?

Code:

A
4                    8
3
x           y
B          M           C
Now, if M is the midpoint between B and C then x and y are equal.
The solution considers them as two different numbers.
AM is the median, right?
How should I proceed now?

Thanks!
1. Draw parallels to AB and AC respectively through M. The parallels divide the sides of the triangle in two equal parts. Together with the known sides of the triangle you'll get a parallelogram with one known diagonal.

2. Use Cosine rule to determine the value of the angle $\alpha$ (see attachment)

3. Determine the value of the angle $\gamma$:

$|\gamma| = 180^\circ - |\alpha|$

4. Use the Cosine rule and the angle $\gamma$ to determine the length of BC. You should come out with $|\overline{BC}|\approx 11.4898$

3. It's also possible to use this formula.

4. Thanks earboth and emakarov!
I searched before posting this question, even in wikipedia.
I guess I overlooked it. It's not so usual to calculate something like this, is it?