More on Angles and Triangles

You are probably begining to realize that angles play an important part in astronomy. Because we shall be using the properties of triangles to help us work with angles, particularly very small angles.
(A) Which of the following are similar triangles? If you multiply the size of all 3 sides of a triangle by the same factor, then the angles remain the same. Similar triangles have the same angles but can be different sizes - scaled up or scaled down.


 

Small angle approximation:

Look at the diagram below.

When d is much smaller than D so that the angle a  is small (say, less than 10°), then we can use the small angle formula: 

a         d
-----  = ------
206,265    D
where a is in arcseconds.


BUT . . . if all we are interested in similar triangles wherea is the same, then we do not need to worry about the number 206,265. More importantly, you can see that for similar triangles:

The following relationship is true:
 
 

d             2d           2.5d          X

---     =     ----     =     ------     =     ---

D            2D           2.5D          Y
 

This means that if we know the values of any 3 of d, D, X and Y, we can work out the 4th.
 
Example: If d=2, D=20, Y=100, then X = Y x d / D = 100 x 2 / 20 = 10.

 
(B) If d=1, D=30, X=5, what is Y? Y = _______________________________________