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A rhombus is formed by joining the midpoints of the sides of a unit square.

What is the diameter of the largest circle that can be inscribed within the rhombus?

  1. $\dfrac{1}{\sqrt{2}}$
  2. $\dfrac{1}{2\sqrt{2}}$
  3. $\sqrt{2}$
  4. $2 \sqrt{2}$
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First, we can draw the diagram.


$\triangle \text{PAQ}$ is a right-angle triangle. So we can apply the Pythagorean theorem.

${\color{Green}{(\text{Hypotenuse})^{2} = (\text{Perpendicular})^{2} + (\text{Base})^{2}}}$

$\Rightarrow (\text{PQ})^{2} = (\text{AQ})^{2} + (\text{AP})^{2}$

$\Rightarrow  (\text{PQ})^{2} = \left(\frac{1}{2}\right)^{2} + \left(\frac{1}{2}\right)^{2}$

$\Rightarrow  \text{PQ} =  \sqrt{\frac{1}{4} + \frac{1}{4}}$

$\Rightarrow  \text{PQ} =  \sqrt{\frac{2}{4}}$

$\Rightarrow  \text{PQ} =  \sqrt{\frac{1}{2}}$

$\Rightarrow {\color{Blue}{\boxed{ \text{XY} =  \frac{1}{\sqrt{2}}}}} \quad [{\color{Red}{\because \text{PQ = XY}}}]$

$\therefore$ The diameter of the largest circle that can be inscribed within the rhombus is $ \dfrac{1}{\sqrt{2}}.$

Correct Answer $:\text{A}$

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