balboa wrote:

Is point \(A\) closer to point \((1, 2)\) than to point \((2, 1)\) ?

1. Point \(A\) lies on the line \(y = x\)

2. Point \(A\) lies on the line \(y = -x\)

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A relevant geometrical property:

A point is equally distant from the endpoints of a given line segment if and only if it belongs to the perpendicular bisector of the line segment.

(You can see it if you draw the perpendicular bisector that each point on it, together with the two endpoints of the line segment, they form an isosceles triangle).

In our case, M(1,2) and N(2,1) define a line segment. The slope of the line going through MN is -1, therefore the line y = x is perpendicular to MN.

The midpoint of MN has coordinates ((1+2)/2,(2+1)/2)=(3/2,3/2) which lies on the line y = x. So, y = x is the perpendicular bisector of MN.

From the above, it follows that Statement (1) is sufficient, as each point on the line y = x is equally distant from M and N.

Statement (2): the line y = -x is parallel to the line segment MN. Except for one point on this line (which also belongs to the line y = x), any point A on the line y = -x will be closer to one of the endpoints of MN, and there are infinitely many points closer to either end of MN. Thus, not sufficient.

Answer A

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