# Total Number of Points

• Oct 14th 2008, 09:04 PM
magentarita
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?
• Oct 14th 2008, 10:12 PM
earboth
Quote:

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.

• Oct 15th 2008, 09:16 AM
masters
Quote:

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.
• Oct 16th 2008, 06:15 PM
magentarita
ok...
Quote:

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?
• Oct 16th 2008, 06:16 PM
magentarita
ok.....
Quote:

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?
• Oct 16th 2008, 07:19 PM
masters
Quote:

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.
• Oct 17th 2008, 02:32 AM
magentarita
but...
Quote:

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.
• Oct 17th 2008, 04:11 AM
masters
Quote:

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.
• Oct 17th 2008, 12:07 PM
magentarita
I see.....
Quote:

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.
• Oct 17th 2008, 01:37 PM
CaptainBlack
Quote:

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