A square is inscribed in a quarter circle in such a way that two of its vertices on the radius are equidistant from the centre and

To find the side length of the square inscribed in a quarter circle, we can solve the problem step-by-step.

1. Break Down the Problem

• Let's denote the radius of the quarter circle as $r$.
• We need to find the side length of the square that fits in the quarter circle such that two of its vertices lie on the radii of the quarter circle.

2. Relevant Concepts

• Consider a square with side length $s$.
• The vertices of the square will touch the circles at some points, while two vertices will lie along the x and y axes.
• The diagonal of the square can be expressed using the Pythagorean theorem since it forms a right triangle with two sides of length $s$.

3. Analysis and Detail

• When the square is inscribed, the coordinates of the corners of the square can be expressed as:
  • Vertex on x-axis: $(s, 0)$
  • Vertex on y-axis: $(0, s)$
• The diagonal length $d$ of the square can be expressed as: $d = s\sqrt{2}$
• This diagonal must also fit within the quarter circle. For the square to be fully inside the quarter circle, it should not exceed the radius, hence: $\sqrt{s^2 + s^2} \leq r$ $s\sqrt{2} \leq r$
• Thus, we find $s$ as follows: $s \leq \frac{r}{\sqrt{2}}$

4. Verify and Summarize

• The maximum side length $s$ of the inscribed square in a quarter circle of radius $r$ is $\frac{r}{\sqrt{2}}$.
• This equation encapsulates the relationship between the square's side length and the radius of the quarter circle.

Final Answer

The side length $s$ of the square inscribed in a quarter circle with radius $r$ is: $s = \frac{r}{\sqrt{2}}$