# Perpendicular lines create a similar triangle?

In my physics book, it says that since both $V_1$ and $V_2$ are perpendicular to the lines joining them to the center of the circle (the blue radius lines), this means that the blue lined triangle (when joined) is a similar triangle to ($V_1 + \Delta V)$, thus making the angle between $V_2, V_1$ being $\theta$.

This is confusing for me, because when searching the internet, it says that there are only two means of a triangle being similar to another, SSS, SAS, and AAA. This text does not meet any of the rules. What am I missing?

• by the way, what do SSS, SAS, AAA mean? I never heard about such acronyms – user126154 May 21 '14 at 8:55
• SSS means Side-Side-Side, i.e. two triangles are "similar" (not equal) when the rispecitve couples of sides are proportional. AAA ia Angle-Angle-Angle, i.e.two triangles are "similar" (not equal) when the rispecitve couples of angles are equal. SAS is Side-Angle-Side, i.e.two triangles are "similar" (not equal) when we consider side $A$ and $A'$ respectively and $B$ and $B'$ and we have that the two couples of proportional and the two angles between $AB$ and $A'B'$ are equal. – Mauro ALLEGRANZA May 21 '14 at 9:02
• See Similarity (geometry) and also this post – Mauro ALLEGRANZA May 21 '14 at 9:09

The angle beween $V_1$ and $V_2$ is $\theta$ because $V_2$ is obtained by a rotation of $V_1$ by $\theta$, as you can check by translating them to the origin ($V_1$ forms an angle $\pi/2$ with the $X$-axis, while $V_2$ $\pi/2+\theta$). If you want to translate this in a language of similiar triangels you can, but personally I see it more clearly by thinking about rotations.

Regarding the similarity between the triangles, the previous answer shows us that the blue triangle formed with the two sides of lenght $r$ and the third one $\Delta r$ [Note : it is not the space travelled on the circular path, but the vector "displacement"], and the other triangle formed by the vectors $V_1$, $V_2$ and $\Delta V$ are similar.

This is so because $|V_1| = |V_2| = r ∗ \omega$, thus :

$$\frac{|V_1|}{|r|}=\frac{|V_2|}{|r|}$$

and the angle formed by $V_1$ and $V_2$ is equal to $\theta$.

Thus we apply the SAS criteria for silmilarity (Side-Angle-Side) and conclude with similarity.

Having shown this, we have that also the third couple of sides must be proportional, i.e. :

$$\frac{|\Delta V|}{|\Delta r|} = \frac{|V_1|}{|r|}=\frac{|V_2|}{|r|}$$

Thus also $|\Delta V|$ and $|\Delta r|$ are proportional with ratio $\omega$ .

• Just a question. How did you get $\Delta V = \Delta s * V / r$ – Jason May 21 '14 at 10:40
• @Jason - You are right : having $\Delta V = \Delta r ∗ \omega$, [with $\Delta r$ and not $\Delta s$], due to the fact that in uniform circular motion : $V = r ∗ \omega$, substituting $V/r$ in place of $\omega$, we get : $\Delta V = \Delta r ∗ V/r$. – Mauro ALLEGRANZA May 22 '14 at 13:22