Ok i really need help with geometry?

Question

ok so can you tell me whether each is true or fault and if its true can you explain why and if it false can you explain to me a counter example

1)if two different lines intersect, then they intersect at one and only one point.

2)if two different circles inersect then they intersect at one and only one point

3)if CA=TA, then a must be the midpoint of line segment CT

4)if point A is not the midpoint of line segment CT then CT does not equal AT

Answer 1

1) true

2) false - If two different circles intersect they intersect at 2 points

3) true

4) true

I'm pretty sure at least. =) but i sucked at math so wait for a few more answers xD

Answer 2

1) True

2) False

3) True

4) False

1) The definition of a straight line in a plane is that two points define exactly one line. If another line could intersect a line in two points, then the lines would have to be the same line:  those two points can define only one line. So if they are different lines, they cannot intersect at two or more points by the definition of "different." (Parallel lines, of course, don't intersect at even one point, but the problem wants to know about different lines that do intersect, not ones that do not. And if the lines can be curved lines, they are not defined this way and could intersect in two to infinity places.)

2) Circles are not defined the same way straight lines are. Specifically, two circles can be different circles even if they share two points. Can they share two points? Yes. (Consider those Olympic rings.) In fact, they can just touch at one point as well. But they cannot share more than two points without being the same circle, not different ones. The Olympic rings are the counter example here.

3) "Line segment" pretty much always means a segment of a straight line. For instance, a segment of a circle's line would be called an arc, not a line segment. We'll go with that. On a line, once points C and A are defined, for point T to be as far from A as C is, point T must be on the other side of A or must be the same point as C. Being the same point as C is not explicitly forbidden here, but common usage absolutely does forbid that (And if so, segment CT then would have zero length and so A would have to be the same point as well and that's just a realllllly screwy excuse to mark your paper wrong!). So we'll go with T being on the far side of A from C. That means A has to line exactly at their segment's middle and that happens to be the definition of a midpoint.

4) I had to say False here because it is not absolute that A lies on segment CT with this wording. In contrast to the certainty of C, T, and A not being the same point in 3). Now, we all know it is supposed to be on segment CT, but it's not absolute that it has to be. If it is somewhere else on line CT than in the segment CT, then if it is on the C side, CT is always smaller than AT and if it is on the T side, it could be far enough that AT is smaller, it could be far enough that AT=CT and it could be so far that AT is greater than CT. We don't know which of those things are true, so False.

Now, in the case that it is on segment CT, like any reasonable person would infer, then clearly, only when it is the same point as C would they equal each other and that's silly, like above, so they cannot equal each other and it would be True, not False.

Sorry to be weaselly about 4). If your teacher is straightforward about things, go with True instead.