1953 AMC 12 Problem 14

Question

Given the larger of two circles with center

P

and radius

p

and the smaller with center

Q

and radius

q.

Draw

\overline{PQ}.

Which of the following statements is false?

p-q

can be equal to

\overline{PQ}

p+q

can be equal to

\overline{PQ}

p+q

can be less than

\overline{PQ}

p-q

can be less than

\overline{PQ}

none of these

Interpret the center distance for internal tangency, external tangency, and disjoint circles

For each of the first four statements, try to choose a valid relative position of the circles

Internal tangency realizes

PQ=p-q,

and external tangency realizes

PQ=p+q.

Disjoint circles can have

PQ\gt p+q,

and many intersecting or disjoint configurations have

PQ\gt p-q.

Thus each of A through D can occur, so none of them is false.

Therefore, the correct answer is E.

Problem 14 in other years