# Thread: Total Number of Points

1. ## Total Number of Points

If point p is on line R, what is the total number of points 3 centimeters from point p and 4 centimeters from line R?

2. Originally Posted by magentarita
If point p is on line R, what is the total number of points 3 centimeters from point p and 4 centimeters from line R?
All points which have a distance of 3 cm from point p are placed on a circle around p with radius r = 3 cm.

All points which have the distance of 4 cm from the line R are placed on 2 parallels of R with a perpendicular distance of 4 cm.

The parallels and the circle don't have common points because the radius is smaller than the perpendicular distance.

3. Originally Posted by earboth
All points which have a distance of 3 cm from point p are placed on a circle around p with radius r = 3 cm.

All points which have the distance of 4 cm from the line R are placed on 2 parallels of R with a perpendicular distance of 4 cm.

The parallels and the circle don't have common points because the radius is smaller than the perpendicular distance.

Actually, if you constructed four 4 cm tangents from line R to the circle, the 4 points of tangency, A, B, C, D would be 4 cm from line R and also 3 cm from P (making 3 - 4 - 5 right triangles). See diagram. Granted, the distances to R are not perpendicular, but the instructions didn't say they had to be. We're just looking for all the points that are both 3 cm from P and 4 cm from line R. So these 4 points meet the conditions.

And as I look further, any point along the arc AB and arc CD with the exception of the intersection points of line R and the circle can be made to be 3 cm from P and 4 cm from line R. So my reasoning may be flawed. Otherwise, there's an infinite number of points that satisfy the condition specified.

4. ## ok...

Originally Posted by earboth
All points which have a distance of 3 cm from point p are placed on a circle around p with radius r = 3 cm.

All points which have the distance of 4 cm from the line R are placed on 2 parallels of R with a perpendicular distance of 4 cm.

The parallels and the circle don't have common points because the radius is smaller than the perpendicular distance.

Why did the other person disagree with you?

5. ## ok.....

Originally Posted by masters
Actually, if you constructed four 4 cm tangents from line R to the circle, the 4 points of tangency, A, B, C, D would be 4 cm from line R and also 3 cm from P (making 3 - 4 - 5 right triangles). See diagram. Granted, the distances to R are not perpendicular, but the instructions didn't say they had to be. We're just looking for all the points that are both 3 cm from P and 4 cm from line R. So these 4 points meet the conditions.

And as I look further, any point along the arc AB and arc CD with the exception of the intersection points of line R and the circle can be made to be 3 cm from P and 4 cm from line R. So my reasoning may be flawed. Otherwise, there's an infinite number of points that satisfy the condition specified.
Are you saying the answer is not zero?

6. Originally Posted by magentarita
Are you saying the answer is not zero?
Earboth's proof is more elequent. I did not take into account that the distances had to be perpendicular distances. So, there is still some question.

7. ## but...

Originally Posted by masters
Earboth's proof is more elequent. I did not take into account that the distances had to be perpendicular distances. So, there is still some question.
This a multiple choice question where zero is one of the choices.

8. Originally Posted by magentarita
This a multiple choice question where zero is one of the choices.
I agree with zero. I just created a situation to fit the problem and I didn't use perpendicular distances. What I demonstrated was the fact that there exists more points that are 3cm from P and also 4 cm from a point on line R, not necessarily perpendicular to line R.

9. ## I see.....

Originally Posted by masters
I agree with zero. I just created a situation to fit the problem and I didn't use perpendicular distances. What I demonstrated was the fact that there exists more points that are 3cm from P and also 4 cm from a point on line R, not necessarily perpendicular to line R.
I now understand.

10. Originally Posted by magentarita
If point p is on line R, what is the total number of points 3 centimeters from point p and 4 centimeters from line R?
There are no such points. Draw a diagram to see why.

CB