Consider a circle inscribed in a square. A rectangle with a base of 2 cm and a height of 1 cm is inserted in the space between one of the vertices of the square and the circumferen... Consider a circle inscribed in a square. A rectangle with a base of 2 cm and a height of 1 cm is inserted in the space between one of the vertices of the square and the circumference in such a way that one vertex of the rectangle coincides with that of the square and the opposite vertex lies on the circumference. Calculate the radius of the circle. Read more

Steps to Solve

1. Visualize the problem Draw a diagram of the square, inscribed circle, and rectangle. Label the radius of the circle as $r$. The center of the circle is at $(r, r)$ if we set the lower left corner of the square as the origin $(0, 0)$.

2. Identify the coordinates of the rectangle's outer vertex The vertex of the rectangle that lies on the circumference of the circle has coordinates $(2, 1)$.

3. Use the circle equation The equation of the circle is $(x - r)^2 + (y - r)^2 = r^2$.

4. Plug in the coordinates of the point (2, 1) Since the point $(2, 1)$ lies on the circle, we have $(2 - r)^2 + (1 - r)^2 = r^2$.

5. Expand and simplify the equation Expanding the equation, we get $4 - 4r + r^2 + 1 - 2r + r^2 = r^2$. Simplifying, we have $r^2 - 6r + 5 = 0$.

6. Solve the quadratic equation We can solve this quadratic equation for $r$. Factoring gives us $(r - 5)(r - 1) = 0$. So, $r = 5$ or $r = 1$.

7. Choose the correct solution Since the rectangle has a base of 2 and a height of 1, the radius must be greater than 2. Therefore, $r = 5$.