1. **State the problem:** Find the unknown segment lengths indicated by "?" in two tangent-secant configurations involving circles. 2. **Recall the tangent-secant theorem:** If a tangent and a secant are drawn from a point outside a circle, then the square of the length of the tangent segment equals the product of the entire secant segment and its external part. Mathematically, if $PT$ is the tangent segment and $PA$ and $PB$ are parts of the secant with $P$ outside the circle, then: $$PT^2 = PA \times PB$$ 3. **Problem 3:** - Given: radius $=4.8$, external segment $=7.2$, unknown segment $=?$ - Let the unknown segment be $x$. - The entire secant segment is $7.2 + x$. - The tangent segment length is the radius $4.8$ (since the tangent touches the circle at radius length). Apply the theorem: $$4.8^2 = 7.2 \times (7.2 + x)$$ Calculate: $$23.04 = 7.2 \times (7.2 + x)$$ Divide both sides by 7.2: $$\frac{23.04}{\cancel{7.2}} = \cancel{7.2} \times \frac{7.2 + x}{\cancel{7.2}}$$ $$3.2 = 7.2 + x$$ Solve for $x$: $$x = 3.2 - 7.2 = -4$$ Since length cannot be negative, re-examine the problem: the tangent length is not the radius but the segment from the external point to the tangent point. The radius is 4.8, but the tangent segment length is unknown. The problem states the radius is 4.8, the external segment is 7.2, and the unknown segment is $x$. Actually, the tangent segment length is unknown, so the tangent length is $x$, and the secant external segment is 7.2, internal segment is 4.8. Apply the theorem: $$x^2 = 7.2 \times (7.2 + 4.8)$$ $$x^2 = 7.2 \times 12 = 86.4$$ Calculate $x$: $$x = \sqrt{86.4} \approx 9.3$$ 4. **Problem 4:** - Given: radius $=7.5$, tangent segment $=17$, unknown segment $=x$ - The secant segment consists of the external part $17$ and internal part $x$. Apply the theorem: $$17^2 = 17 \times (17 + x)$$ Calculate: $$289 = 17 \times (17 + x)$$ Divide both sides by 17: $$\frac{289}{\cancel{17}} = \cancel{17} \times \frac{17 + x}{\cancel{17}}$$ $$17 = 17 + x$$ Solve for $x$: $$x = 17 - 17 = 0$$ This suggests the unknown segment is zero, which is not possible. Re-examine: the tangent segment length is 17, the radius is 7.5, and the unknown segment is inside the circle. The tangent length squared equals the product of the entire secant segment and its external part: $$17^2 = (7.5 + x) \times x$$ Calculate: $$289 = (7.5 + x) x = 7.5x + x^2$$ Rewrite: $$x^2 + 7.5x - 289 = 0$$ Solve quadratic equation: $$x = \frac{-7.5 \pm \sqrt{7.5^2 + 4 \times 289}}{2} = \frac{-7.5 \pm \sqrt{56.25 + 1156}}{2} = \frac{-7.5 \pm \sqrt{1212.25}}{2}$$ Calculate square root: $$\sqrt{1212.25} = 34.83$$ Two solutions: $$x = \frac{-7.5 + 34.83}{2} = \frac{27.33}{2} = 13.67$$ $$x = \frac{-7.5 - 34.83}{2} = \frac{-42.33}{2} = -21.17$$ Length cannot be negative, so: $$x = 13.67$$ **Final answers:** - Problem 3: $x \approx 9.3$ - Problem 4: $x \approx 13.67$