A line 'l' intersects line 'm' at point A at 45 degree angle. B and C are points on line m, where the distance from B to A is equal to the distance from A to C. How many points on line 'l' are equidistant from points B and C?

How is it one point? And this one point on line 'l' is the point that intersects point A? I thought that points B and C are the same points, so if I drew a vertical line perpendicular to line 'm' then as long as the horizontal distance from point B or C to this other new point is equal, then can't there be an infinite amount of points? Because this new point can be anywhere on line 'l'?

Let me know your explanation because I think I may have not understood the question or drawn the diagram correctly.

Thanks!

I assume point A is on line l

Place B and C anywhere on line m
draw the perpendicular bisector of BC, and extend the line
Notice that no matter where you draw B and C, this right bisector will cut line l only once.
label that point A.
properties of this right bisector:
AB = AC

In order to understand the problem correctly, let's go step by step.

1. Line l intersects line m at point A at a 45-degree angle.
This means that lines l and m meet at a point A, and the angle formed at this intersection is 45 degrees.

2. Points B and C are on line m, where the distance from B to A is equal to the distance from A to C.
This means that the lengths of line segments BA and AC are equal.

Now, let's address your concerns:

You mentioned that points B and C are the same points. However, in the problem statement, it is stated that B and C are separate points on line m. The distance from B to A is equal to the distance from A to C. So, we have a segment BA and segment AC, which have the same length.

The question asks how many points on line l are equidistant from points B and C.

To visualize this, you can draw a diagram as follows:

- Draw line m.
- Mark point A on line m as the intersection with line l.
- Mark points B and C on line m such that the distance from B to A is equal to the distance from A to C.

Now, we need to find the points on line l that are equidistant from points B and C.

To do this, draw a circle centered at point B with radius BA (the length of segment BA). Similarly, draw another circle centered at point C with radius CA (the length of segment CA).

The intersection point of these two circles will be the point on line l that is equidistant from points B and C. This is because any point on the circumference of a circle is equidistant from its center.

Therefore, there is only one point on line l that satisfies the condition of being equidistant from points B and C, and it is the point where the two circles intersect.

Note: If the two circles do not intersect, it means that there is no point on line l that is equidistant from points B and C.

I hope this explanation clarifies the situation for you. If you have any further questions, please let me know!