1. **State the problem:** Find the unknown segment lengths $x$ using the appropriate circle theorems. --- ### Part I: Determine the theorem to use 1. We have a tangent segment of length 15 and a secant with external segment $x$ and internal segment 14. **Theorem:** Tangent-Secant Theorem states: $$\text{tangent}^2 = \text{external secant segment} \times \text{whole secant segment}$$ 2. Two secants intersecting inside the circle with segments 9 and $x$ on one, and 16 and 18 on the other. **Theorem:** Chords/Intersecting Secants Inside Theorem: $$\text{product of segments on one chord} = \text{product on the other}$$ 3. Two secants intersecting outside the circle with external segments 18 and $x$, internal segments 22 and 29 respectively. **Theorem:** Two Secants Theorem (intersecting outside): $$\text{external segment} \times \text{whole secant} = \text{external segment} \times \text{whole secant}$$ --- ### Part II: Solve for $x$ and segment measures 1. - Set-up (tangent-secant): $$15^2 = x(x + 14)$$ - Compute: $$225 = x^2 + 14x$$ - Rearrange: $$x^2 + 14x - 225 = 0$$ - Solve quadratic: $$x = \frac{-14 \pm \sqrt{14^2 + 4\times 225}}{2} = \frac{-14 \pm \sqrt{196 + 900}}{2} = \frac{-14 \pm \sqrt{1096}}{2}$$ Since length must be positive, $$x = \frac{-14 + 33.1}{2} = 9.55$$ (approx) - Measures: $$QA = x = 9.55$$ $$AC = QA + 14 = 9.55 + 14 = 23.55$$ $$DB = 15 \ (tangent)$$ 2. - Given secants intersecting inside, so: $$(9)(x) = (16)(18)$$ - Solve: $$9x = 288$$ $$x = 32$$ - Measures: $$EF = x + 9 = 32 + 9 = 41$$ (if $EF$ is whole secant) $$GE = 16 + 18 = 34$$ (Problem statement incomplete for this part; assuming we only needed $x$.) 3. - For outside secants: $$(18)(18+22) = x (x + 29)$$ - Compute: $$18 \times 40 = x (x + 29)$$ $$720 = x^2 + 29x$$ - Rearrange: $$x^2 + 29x - 720 = 0$$ - Solve quadratic: $$x = \frac{-29 \pm \sqrt{29^2 + 4 \times 720}}{2} = \frac{-29 \pm \sqrt{841 + 2880}}{2} = \frac{-29 \pm \sqrt{3721}}{2}$$ $$\sqrt{3721} = 61$$ Choose positive solution: $$x = \frac{-29 + 61}{2} = 16$$ - Measures: $$LM = 2x + 3 = 2(16) + 3 = 35$$ $$KM, KU$$ measures depend on specific segments from problem (not fully provided here).