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GRE Quantitative Reasoning QUESTION #8093
Question 1

Given the diagram of three circles with centers $P$, $Q$, and $R$ forming an equilateral triangle $PQR$, where each pair of circles has exactly one point in common:

QPR

Compare:

Column A: The perimeter of triangle $PQR$

Column B: The circumference of the circle with center $Q$

  • Column A is greater

  • Column B is greater

    ✔️

  • The two quantities are equal

  • The relationship cannot be determined

Correct Answer Explanation

Since each pair of circles has exactly one point in common, the circles are externally tangent to each other. If all circles have the same radius $r$, then the distance between any two centers equals $2r$.

Since triangle $PQR$ is equilateral with side length $2r$:

  • Perimeter of $\triangle PQR = 3(2r) = 6r$
  • Circumference of circle with center $Q = 2\pi r$

Comparing: $6r$ vs $2\pi r$

Dividing by $2r$: $3$ vs $\pi$

Since $\pi \approx 3.14159 > 3$, we have $2\pi r > 6r$.

Therefore, Column B (the circumference) is greater than Column A (the perimeter).