1. Let's analyze the left diagram first. According to the intersecting chords and secants properties in a circle: For a point outside the circle where a secant intersects the circle, the product of the external segment and the whole secant segment equals the product of the segments of the chord inside. 2. From the left diagram, we have: - External part of the secant: 10 - Whole secant length: 10 + 5 = 15 - Chord segments (diameter divided into two equal parts since it's a diameter): each segment is $x$ - Diameter length: $2x$ Using the secant-tangent product theorem, we get: $$10 \times 15 = x \times 2x$$ 3. Simplify the left-hand side: $$150 = 2x^2$$ 4. Solve for $x^2$: $$x^2 = \frac{150}{2} = 75$$ 5. Find $x$: $$x = \sqrt{75} = 5\sqrt{3}$$ 6. Now let's examine the right diagram. The external segments connected outside the circle are 5 and 4, and the segment inside the circle is $x$. By the Power of a Point theorem for two secants: $$5 \times (5 + 4) = x \times (x + 4)$$ 7. Simplify the left-hand side: $$5 \times 9 = 45$$ 8. Expand the right-hand side: $$x (x + 4) = x^2 + 4x$$ 9. Set up the equation: $$x^2 + 4x = 45$$ 10. Rearrange to standard quadratic form: $$x^2 + 4x - 45 = 0$$ 11. Solve quadratic using the quadratic formula where $a=1$, $b=4$, and $c=-45$: $$x = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times (-45)}}{2 \times 1} = \frac{-4 \pm \sqrt{16 + 180}}{2} = \frac{-4 \pm \sqrt{196}}{2}$$ 12. Simplify the square root: $$x = \frac{-4 \pm 14}{2}$$ 13. Find the two solutions: - $$x = \frac{-4 + 14}{2} = \frac{10}{2} = 5$$ - $$x = \frac{-4 - 14}{2} = \frac{-18}{2} = -9$$ Since length cannot be negative, discard $x = -9$. 14. Final solutions: - For the left diagram, $x = 5\sqrt{3}$ - For the right diagram, $x = 5$