tangent

[shahar]

How I prove the underline text:
A line that touch only with one point a circle called tangent. By the symmetry property of the circle we can show that the tangent is normal (vertical) to radius in the point of tangent.
What are the meaning of the property of symmetry and the connection to the tangential line?

 

[pka]

How I prove the underline text: A line that touch only with one point a circle called tangent. By the symmetry property of the circle we can show that the tangent is normal (vertical) to radius in the point of tangent. What are the meaning of the property of symmetry and the connection to the tangential line?

Frankly, I think only the person who wrote this knows for for sure.
I can explain why it is true. The line segment from the center of a circle to a point of the circle(known as a radial segment) has length equal to the radius of the circle. Because the tangent line contains only one point of the circle, that is the point on the tangent that is closest to the center. But we can prove that is the perpendicular distance. Thus the radial segment is perpendicular to the tangent at the point of contact with the circle.
That, I realize, is a bit convoluted but I hope it helps.

 

[HallsofIvy]

I would take the "symmetry property of the circle" to be the fact that a diameter divided the circle into two identical semi-circles. So a radius (part of a diameter) meets a tangent symmetrically- the angles on either side of the radius are equal so 180/2= 90 degrees.

 

[shahar]

I would take the "symmetry property of the circle" to be the fact that a diameter divided the circle into two identical semi-circles. So a radius (part of a diameter) meets a tangent symmetrically- the angles on either side of the radius are equal so 180/2= 90 degrees.

I can't picture the underlined text, can you show a picture of that.

 

[Steven G]

No picture needed. Here is what is going on. From a point outside a circle draw a straight line that touches the circle in one place (there are two such places). Call the initial point P and the point of tangent C.

Now from point C draw a line to the center of the circle. These two lines are perpendicular

 

[Dr.Peterson]

How I prove the underline text:
A line that touch only with one point a circle called tangent. By the symmetry property of the circle we can show that the tangent is normal (vertical) to radius in the point of tangent.
What are the meaning of the property of symmetry and the connection to the tangential line?

There are probably many ways you could use symmetry in such a proof; I'm not sure what the author specifically had in mind. Halls' idea, I think, is that if you reflect the entire figure (circle and tangent line) over the line through the center O and the point of tangency C, it must remain unchanged (that's what symmetry means); so the angles on each side at C must be equal. That implies they are both right angles.

You could also approach this as a proof by contradiction, by supposing that the line is not perpendicular to the radius, doing the reflection, and showing that the line would not be tangent (because it would have to intersect the circle in two points).

If you want help figuring out what proof the author was thinking of, perhaps that would be clearer if we saw more of the context, including the explicit statement of the symmetry property of the circle, and any other examples of how the author uses symmetry. That's what I'd be looking at if I were you.

Incidentally, I would say "the tangent is perpendicular to the radius"; the word "normal" is valid, but I'd use it in different contexts, while "vertical" is not appropriate here.