Home › MCQs › GRE Quantitative Reasoning › Question #8095
Back to Questions
GRE Quantitative Reasoning QUESTION #8095
Question 1

In the figure above, each of the four squares has sides of length $x$. If $\triangle PQR$ is formed by joining the centers of three of the squares, what is the perimeter of $\triangle PQR$ in terms of $x$?

PQR

  • $2x\sqrt{2}$

  • $\dfrac{x\sqrt{2}}{2}+x$

  • $2x+x\sqrt{2}$

    ✔️

  • $x\sqrt{2}+2$

Correct Answer Explanation

Each square has side length $x$, so the center of each square is at distance $\frac{x}{2}$ from each edge.

From the arrangement:

  • $P$ is the center of the top-left square at $\left(\frac{x}{2}, \frac{x}{2}\right)$
  • $Q$ is the center of the top-right square at $\left(\frac{3x}{2}, \frac{x}{2}\right)$
  • $R$ is the center of the bottom-right square at $\left(\frac{3x}{2}, \frac{3x}{2}\right)$

Distances:

  • $PQ = x$ (horizontal distance between adjacent centers)
  • $QR = x$ (vertical distance between adjacent centers)
  • $PR = \sqrt{x^2 + x^2} = x\sqrt{2}$ (diagonal)

Perimeter $= PQ + QR + PR = x + x + x\sqrt{2} = 2x + x\sqrt{2}$