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why do the lines, selected above, appear to be tangent lines, and why are the other line(s) not tangent lines?

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Final answer:

Tangent lines touch a curve at exactly one point without cutting through it. Whereas lines that aren't tangents either intersect at multiple points or do not touch the curve at all.

Step-by-step explanation:

When discussing tangent lines, we are referring to a line that touches a curve at a single point without crossing over it. A tangent line to a circle, for instance, intersects the circle's circumference at exactly one point. In the context of your question, the lines that are considered to be tangent lines meet this criterion. They 'kiss' the curve at one point only.

Other lines that are not considered tangent lines might cross the curve at more than one point or not touch the curve at all. These could be secant lines, which intersect the curve at two points, or external lines, which do not meet the curve at any point. For example, if we have a circle and a line that touches it at exactly one point without cutting through the circle, this line is called a tangent line. Any line that intersects the circle at two points or does not touch it at all is not a tangent.

User Andrew Luo
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