# Trigonometry, knowing 3 sides how to find the height?

I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is. What I realy want to find is the height of the triangle and another one "side". Let me explain what I want, with the above picture. I knew a,b and c and I can calculate the 2 angles (the angle oposite of C, and the angle oposite of b, those on the black dots). I want to find out the red dots (the bottom red dot is is the other "side" I note before, from black dot to red dot distance).

So I want to to know the (x,y) for the red dot at the top, via the combination of height length, b OR c length (which also needs the two angles) in case to find the absolute right (x,y) for the red dot at the top. Thanks.

Use Heron's Formula for the area $A$ of a triangle

$$A=\sqrt{p\left( p-a\right) \left( p-b\right) \left( p-c\right) }=\frac{a\cdot h}{2},\tag{1}$$

where $h$ is the altitude length and $$2p=a+b+c\tag{2}$$ is the perimeter. A geometric proof of $(1)$ can be found in this post, in Portuguese. Solving $(1)$ for $h$ we find that

\begin{eqnarray*} h &=&\frac{2\sqrt{p\left( p-a\right) \left( p-b\right) \left( p-c\right) }}{a } \\ &=&\frac{2}{a}\sqrt{\frac{a+b+c}{2}\left( \frac{a+b+c}{2}-a\right) \left( \frac{a+b+c}{2}-b\right) \left( \frac{a+b+c}{2}-c\right) } \\ &=&\frac{1}{2a}\sqrt{\left( a+b+c\right) \left( b+c-a\right) \left( a+c-b\right) \left( a+b-c\right) }\tag{3} \end{eqnarray*}

how this will help me to find the (x,y) for the top red dot in the picture?

Concerning coordinates, if the base $a$ is horizontal and the coordinates of the left vertex are $(x_{B},y_{B})=(0,0)$, then the coordinates of the upper vertex are

$$(x_{A},y_{A})=(\sqrt{c^{2}-h^{2}},h).\tag{4}$$

The coordinates of the foot of the altitude $h$ are thus $$(x_{N},y_{N})=(\sqrt{c^{2}-h^{2}},0).\tag{5}$$ • thanks, this will give me the area, but how this will help me to find the (x,y) for the top red dot in the picture? – Dr. Mathematician Feb 18 '14 at 1:39
• @ProgrJohn This will give you the height $h$. – Américo Tavares Feb 18 '14 at 1:41
• thanks man. I think this will work, I did some paper calculations and it is works. I will bring the solution to my mathematician tutor. – Dr. Mathematician Feb 18 '14 at 13:10
• @ProgrJohn You are welcome. Glad to help. – Américo Tavares Feb 18 '14 at 13:11

Or from Heron's formula for triangle area http://en.wikipedia.org/wiki/Heron%27s_formula

• Algebraic proof using the Pythagorean theorem? this will work for any kind of triangle? – Dr. Mathematician Feb 18 '14 at 1:34
• Heron is completely general. There are a couple trig proofs and one algebraic in the link. – ir7 Feb 18 '14 at 1:36

Because you know $b$ and $c$ and the two lower angles, you can use cosine to find the lengths of the parts of the bottom side, which gives you the lower dot. You can use sine to find the height of the triangle.