Problem 9 · The staircase inside the quarter circle
Two points on the arc and a system solved by subtracting.
Integer answer, at most 4 digitsThe figure shows a quarter circle with some measurements and right angles marked. What is the squared radius ($r^{2}$) of the quarter circle? If the result is a fraction, simplify it fully and give the sum of numerator and denominator.
Reasoned solution
Key idea: place the centre at the origin. The staircase leaves the vertical side, and its two outer elbows lie on the arc.
The first leg (9 cm) goes from the vertical side to the point $A = (9,\ y_0)$ on the arc; we go down $16$ and turn: the end of the 12 cm leg is $B = (21,\ y_0 - 16)$, also on the arc. The two conditions:
Subtracting: $441 - 81 - 32y_0 + 256 = 0 \Rightarrow 32y_0 = 616 \Rightarrow y_0 = \dfrac{77}{4}$.
Numerator plus denominator: $7225 + 16 = 7241$.