# Is this true for all tangents to a circle?

#### StephenPoco

Well, I was testing somethings out to try and find a particular point that is a tangent to a circle and I decided to try this. I'm not too sure whether it's right or not.

I just want to know if this is true for all tangents to a circle? Excuse the terrible picture. I just drew it in a rush on MS Paint. (Rofl) I am hoping you can decipher it.

#### earboth

MHF Hall of Honor
Well, I was testing somethings out to try and find a particular point that is a tangent to a circle and I decided to try this. I'm not too sure whether it's right or not.

I just want to know if this is true for all tangents to a circle?

Excuse the terrible picture. I just drew it in a rush on MS Paint. (Rofl) I am hoping you can decipher it.
You've found a very special case which is only true if $$\displaystyle |\overline{CR}| = |\overline{CQ}|$$

In general:
1. Use the indicated blue triangle to determine the slope of the tangent:

$$\displaystyle m_t=-\frac1{m_{radius}}~\implies~m_t=-\frac{X_P - X_C}{Y_P - Y_C}$$

2. Use the triangle RCQ to determine the slope of the tangent:

$$\displaystyle m_t=\frac{Y_Q-Y_R}{X_Q-X_R}$$

Since $$\displaystyle X_R = X_C$$ and $$\displaystyle Y_Q = 0$$ the 2nd equation becomes:

$$\displaystyle m_t=\frac{-Y_R}{X_Q-X_C}$$

3. You now can use these 2 equations to get some additional results.

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