Intersecting chords

Problem

Two chords $AB$ and $CD$ intersect at a point $S$ inside the circle. Knowing that $\begin{cases} \overline{AB} = 5 \\ SC = \tfrac{1}{3}\,SB \\ DS = 3 \end{cases}$ determine $SA$, $SB$ and $SC$.

Solution

Set $SB = x$, so $SC = \dfrac{x}{3}$ and, since $SA + SB = \overline{AB} = 5$, we have $SA = 5-x$.

Apply the intersecting chords theorem $SA\cdot SB = SC\cdot SD$: $x\cdot\frac{x}{3} = 3\cdot(5-x) \implies \frac{x^2}{3} = 15-3x.$ Multiplying by $3$ and rearranging: $x^2 + 9x - 45 = 0 \implies x = \frac{-9+\sqrt{81+180}}{2} = \frac{-9+\sqrt{261}}{2} \approx 3{,}58.$ (the negative root is discarded because $x=SB>0$). Hence: $SB = \boxed{\tfrac{-9+\sqrt{261}}{2}\approx 3{,}58}, \quad SA = 5-x \approx \boxed{1{,}42}, \quad SC = \frac{x}{3} \approx \boxed{1{,}19}.$

Links

Topics: Euclidean circle
Concepts: Chord · Power of a point
Methods: Circle chords
Skills: Synthetic geometry · Solving equations
Exercise type: Geometric problem