# identify tangent line

• Jul 20th 2012, 07:04 AM
identify tangent line
Attachment 24319 there are two lines AB and CD which one is tangent line and why?

I know they say AB is tangent line but how to Mathematically( Geometrically(Trigonometrically )) prove it?

we can take tangent theta too and tangent psi too then both are through tangent of some angle so aren't they both tangent lines?
• Jul 20th 2012, 08:13 AM
ILikeSerena
Re: identify tangent line
The curve is smooth and there is only one line that is (locally) entirely on one side of the curve.
• Jul 21st 2012, 09:30 AM
Re: identify tangent line
Good answer to some extent but why isn't CD a tangent line?
• Jul 21st 2012, 09:38 AM
HallsofIvy
Re: identify tangent line
CD is not going in the same direction as the curve at the point where they cross.
• Jul 21st 2012, 09:39 AM
astartleddeer
Re: identify tangent line
Quote:

Good answer to some extent but why isn't CD a tangent line?

I'd say, if you were to extend $y= f(x)$ and also the line segment CD. Then the line segment p to D becomes the secant, from which will give the average change until $\lim_{x \to 0}$ the secant becomes the tangent at point p.
• Jul 21st 2012, 09:40 AM
ILikeSerena
Re: identify tangent line
Ah well, let's see what wiki says:

Quote:

Originally Posted by wiki
In geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point—that is, coincides with the curve at that point and, near that point, is closer to the curve that any other line passing through that point. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f.

As you can see, CD does not "just touch".

And if we look at the derivative, we'd be looking at a line through P and a second point on the curve.
When we let the second point approach P, we get the tangent line.
This is AB and not CD.
• Jul 21st 2012, 09:46 AM
HallsofIvy
Re: identify tangent line
Quote:

we can take tangent theta too and tangent psi too then both are through tangent of some angle so aren't they both tangent lines?
You need to review the definition of "tangent line". It has nothing to do with "tangent of an angle".