# Geometry, Find sides of a triangle

I have this equilateral triangle. The sun shaped object is a sound source. I know the difference of time between the arrival of the sound at the 2 vertices of the triangle. I need to find the angle alpha based on this information. I know the length of the side of the equilateral triangle.

Any help will be appreciated.

• You know the lengths from the sound source to the two vertices, and you know the length of a side of the equilateral triangle. Try using the law of cosines to fill in more unknown angles. – user137794 Apr 15 '14 at 22:18
• What's the length you know (assuming that's what you meant by difference of time)? – Shahar Apr 15 '14 at 22:19
• I do not know the lengths from the source. All I know is at which vertex the sound reaches first and what is the difference in time between the arrival of sound at the 2 vertices. – user2731223 Apr 15 '14 at 22:25
• They don't have to be similar, all they need to satisfy is that the angle alpha remains the same in all these 3 cases. – user2731223 Apr 15 '14 at 23:42
• You haven't said this, but is the vertex of the angle $\alpha$ supposed to be the center of the equilateral triangle? – Gerry Myerson Apr 16 '14 at 7:00

Different positions on the upper branch of the hyperbola would lead to different angles $\alpha$. So the angle isn't known either. In many situations, the most reasonable thing you can do is concentrate on the case where the distance between sensors and sound source is much larger than the distance between the two sensors. So you could use the direction of the asymptote as an approximation.