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[tex]$\overrightarrow{AB}$[/tex] lies in the same plane as [tex]$\odot P$[/tex], which has a radius of [tex]$6$[/tex] mm. At how many points can [tex]$\overrightarrow{AB}$[/tex] intersect [tex]$\odot P$[/tex] if [tex]$AP = 6$[/tex] mm and [tex]$BP = 8$[/tex] mm?

A. 1 point or 2 points
B. 0 points or 1 point
C. 0 points or 2 points
D. There is not enough information to determine.

Sagot :

GinnyAnswer
Let's analyze the situation step-by-step:

1. Understanding the Circle and Points:
- We have a circle [tex]\( \odot P \)[/tex] with center [tex]\( P \)[/tex] and radius [tex]\( 6 \)[/tex] mm.
- Point [tex]\( A \)[/tex] is given such that [tex]\( AP = 6 \)[/tex] mm. This implies point [tex]\( A \)[/tex] is on the circle since the distance [tex]\( AP \)[/tex] is exactly equal to the radius of the circle.

2. Determining the Positions of Points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Point [tex]\( B \)[/tex] is given such that [tex]\( BP = 8 \)[/tex] mm.
- Since [tex]\( BP \)[/tex] is greater than the radius of the circle, point [tex]\( B \)[/tex] lies outside the circle.

3. Intersection Analysis Between Line Segment [tex]\( \overrightarrow{AB} \)[/tex] and the Circle [tex]\( \odot P \)[/tex]:
- Since point [tex]\( A \)[/tex] is on the circle and point [tex]\( B \)[/tex] is outside the circle, we need to evaluate how many points the line segment [tex]\( \overrightarrow{AB} \)[/tex] intersects the circle.

4. Checking Intersection Points:
- When one endpoint of the line segment (point [tex]\( A \)[/tex]) is on the circle and the other endpoint (point [tex]\( B \)[/tex]) is outside the circle, the line segment [tex]\( \overrightarrow{AB} \)[/tex] must intersect the circumference of the circle at two distinct points because it must enter and exit the circle.
- Hence, point [tex]\( A \)[/tex] provides one intersection point (since it lies on the circle), and the segment will re-enter the circle at another distinct point before reaching point [tex]\( B \)[/tex]. This confirms two intersection points overall.

### Conclusion

From the above analysis:

- The line segment [tex]\( \overrightarrow{AB} \)[/tex] intersects the circle [tex]\( \odot P \)[/tex] at exactly 2 points.

Thus, the correct answer is 2 points, and the best fitting answer from the given choices is:

A. 1 point or 2 points

Given the specific conditions, the answer more accurately aligns with this option as two points of intersection are consistent with such scenarios.
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