1. Let's start by stating the problem: We need to create and analyze three types of circle problems involving secants and tangents: intersecting circles, secant-secant, and secant-tangent. 2. **Intersecting Circles Problem:** Given two circles that intersect at points $A$ and $B$, find the length of the chord $AB$ if the radii and distance between centers are known. 3. **Secant-Secant Theorem:** If two secants are drawn from a point outside a circle, intersecting the circle at points $A$, $B$ and $C$, $D$ respectively, then the products of the lengths of the segments are equal: $$PA \times PB = PC \times PD$$ 4. **Secant-Tangent Theorem:** If a secant and a tangent are drawn from a point $P$ outside the circle, touching the circle at $T$ (tangent point) and intersecting at $A$ and $B$ (secant points), then: $$PT^2 = PA \times PB$$ 5. **Example Problem 1 (Intersecting Circles):** Two circles with radii 5 and 3 units have centers 7 units apart. Find the length of chord $AB$ where they intersect. 6. **Solution:** Using the intersecting chord theorem and triangle properties, the length of chord $AB$ is found by: $$AB = 2 \sqrt{r_1^2 - d^2/4}$$ where $r_1=5$ and $d=7$. Calculate: $$AB = 2 \sqrt{5^2 - (7^2)/4} = 2 \sqrt{25 - 49/4} = 2 \sqrt{\frac{100 - 49}{4}} = 2 \times \frac{\sqrt{51}}{2} = \sqrt{51}$$ So, $AB = \sqrt{51}$ units. 7. **Example Problem 2 (Secant-Secant):** From point $P$ outside a circle, two secants intersect the circle at $A$, $B$ and $C$, $D$ such that $PA=4$, $PB=6$, and $PC=3$. Find $PD$. 8. **Solution:** Using the secant-secant theorem: $$PA \times PB = PC \times PD$$ $$4 \times 6 = 3 \times PD$$ $$24 = 3 \times PD$$ $$PD = 8$$ 9. **Example Problem 3 (Secant-Tangent):** From point $P$ outside a circle, a tangent touches the circle at $T$ and a secant intersects at $A$ and $B$ with $PA=5$, $PB=9$. Find $PT$. 10. **Solution:** Using the secant-tangent theorem: $$PT^2 = PA \times PB = 5 \times 9 = 45$$ $$PT = \sqrt{45} = 3\sqrt{5}$$ **Final answers:** - Length of chord $AB$ in intersecting circles: $\sqrt{51}$ units - Length $PD$ in secant-secant: 8 units - Length $PT$ in secant-tangent: $3\sqrt{5}$ units