# Thread: Urgent! How to find inner angles and side of the rhombus given height and diagonal e?

1. ## Urgent! How to find inner angles and side of the rhombus given height and diagonal e?

We are given in rhombus that height is equal $h = 2\sqrt{3} cm$ and diagonal $AC = 4\sqrt{3} cm$. Find length of a rhombus side $a$, other diagonal $BD$ all inner angles.

2. ## Re: Urgent! How to find inner angles and side of the rhombus given height and diagona

And do you know what a rhombus is?

3. ## Re: Urgent! How to find inner angles and side of the rhombus given height and diagona

Originally Posted by lebdim
We are given in rhombus that height is equal $h = 2\sqrt{3} cm$ and diagonal $AC = 4\sqrt{3} cm$. Find length of a rhombus side $a$, other diagonal $BD$ all inner angles.
something like this?

4. ## Re: Urgent! How to find inner angles and side of the rhombus given height and diagona

Nice picture! Calling the vertex at (0,0) "A" and going around counter clockwise, labeling vertices A, B. C, D, then $a= h cos(\theta)= 2\sqrt{3}cos(\theta)$ where $\theta$ is angle BAD. Further, triangle ABC is an isosceles triangle with base angles $\theta/2$ and two sides of length a. By the "cosine law", the length of the third side, the diagonal AC is given by $|AC|^2= (4\sqrt{3})^2= 48= a^2+ a^2+ 2a^2 cos(\theta/2)$. That gives two equations,
$a= 2\sqrt{3}cos(\theta)$ and $(4\sqrt{3})^2= 48= 2a^2(1+ a^2 cos(\theta/2))$ to solve for a and $\theta$.

5. ## Re: Urgent! How to find inner angles and side of the rhombus given height and diagona

The coordinates are to scale ... all right triangles are 30-60-90.

6. ## Re: Urgent! How to find inner angles and side of the rhombus given height and diagona

@HallsofIvy,

I have a question. How can be the side $a = h\cos{\theta}$, if $a$ is a hypotenusis? Isn't it $\sin{\theta} = \frac{h}{a}$?

And cosine-law, I got: $a^{2} = a^{2} + e^{2} - 2\cdot{a}\cdot{e}\cdot\cos{\frac{\theta}{2}}$? Made I mistake, where $e = \left|AC\right|$?