Problem 13 · The semicircle with three segments
Two arc points and an unknown centre: two circle equations.
Integer answer, at most 4 digitsIn the semicircle of this figure some segments have been drawn. If $\overline{AB} = 7$ cm, $\overline{CD} = 6$ cm and $\overline{DE} = 5$ cm, what is the area of the semicircle? The answer is a multiple of $\pi$: give that number without $\pi$ (if the answer is $40\pi$, answer $40$; if it's decimal, round to the nearest integer).
Reasoned solution
Key idea: set coordinates with the diameter on the $x$-axis: $B$ and $D$ lie on the arc and their heights and abscissas can be read off the figure.
Let $A = (t, 0)$. Then $B = (t, 7)$ (since $AB \perp$ diameter), $C = (t, 5)$ and $D = (t+6, 5)$ (since $CD$ is horizontal with $DE = 5$ vertical). With centre $O = (c, 0)$ and radius $r$:
Subtracting: $\bigl(2(t-c)+6\bigr) \cdot 6 = 24$, i.e. $t - c = -1$, hence $r^2 = 1 + 49 = 50$.