In the figure below, the rectangle at the corner measures 10 cm \( \times \) 20 cm. The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?
Show Hint
Put the centre at distance r from two sides of the square and write the distance from the centre to A. Both roots of the resulting quadratic need to be tested against the picture.
Step 1: Understanding the Concept:
The circle sits inside a square and touches its sides. Its centre is therefore at the same distance r from the left side and from the top side, where r is the radius. If we place the top-left corner of the square at the origin, the centre is at \( (r, r) \). The small rectangle is 20 cm wide and 10 cm tall, so its far corner A is at \( (20, 10) \) when we measure x to the right and y downward.
Step 2: Key Formula or Approach:
A point lies on a circle when its distance from the centre equals the radius. So the distance from \( (r, r) \) to \( (20, 10) \) must be r.
Step 4: Solve the quadratic.
Two numbers that multiply to 500 and add to 60 are 10 and 50. So \[ (r - 10)(r - 50) = 0 \Rightarrow r = 10 \text{ or } r = 50 \]
Step 5: Pick the root that fits the figure.
If r = 10, the square would be only 20 cm by 20 cm. The 20 cm wide rectangle would then run across the whole top edge, and A would be the point where the circle touches the right side. The figure clearly shows a small rectangle in the corner and the circle much bigger than it. So r = 10 is rejected and r = 50 is the radius.
Step 6: Check the options.
Option 1 (10 cm) is the root we just rejected. Option 2 (40 cm) fails the equation: \( 20^2 + 30^2 = 1300 \) but \( 40^2 = 1600 \). Option 4 is wrong because 50 cm is listed. Option 3 (50 cm) works: \( 30^2 + 40^2 = 2500 = 50^2 \).
Final Answer:
The radius of the circle is 50 cm, which is option 3.
\[ \boxed{50 \text{ cm}} \]
