1. **Problem 1:** Given a triangle inside a circle with sides 8.8, x, and radius 22.4, find the value of $x$ from options A) 5, B) 4.3, C) 4.4, D) 3.1. 2. **Step 1:** Use the Pythagorean theorem if the triangle is right-angled (common in circle radius problems): $$x^2 + 8.8^2 = 22.4^2$$ 3. **Step 2:** Calculate squares: $$8.8^2 = 77.44$$ and $$22.4^2 = 501.76$$ 4. **Step 3:** Substitute and solve for $x$: $$x^2 = 501.76 - 77.44 = 424.32$$ 5. **Step 4:** Find $x$: $$x = \sqrt{424.32} \approx 20.6$$ which is not among the options, so check if the problem implies a different approach or if $x$ is a chord length or segment. 6. **Problem 2:** Given circle $O$ with chords $CD$ and $AB$, and angles $\angle AC$ and $\angle DB$, determine which statements are true: - i. $CD = AB$ - ii. $m\angle AC = m\angle DB$ - iii. $m\angle DC = m\angle AB$ 7. **Step 1:** In a circle, equal chords subtend equal angles at the center and are equal in length. 8. **Step 2:** If $CD = AB$ (i), then angles subtended by these chords are equal (ii). 9. **Step 3:** Angles $m\angle DC$ and $m\angle AB$ are not necessarily equal unless given. 10. **Answer:** Statements i and ii are true. 11. **Problem 3:** In circle $C$, chords $LM$ and $NP$ intersect. Given: $$LM = 5x + 4$$ $$NP = 9x - 12$$ $$CN = 13$$ Find $CY$. 12. **Step 1:** Since $LM = NP$, set equal: $$5x + 4 = 9x - 12$$ 13. **Step 2:** Solve for $x$: $$5x + 4 = 9x - 12 \Rightarrow 4 + 12 = 9x - 5x \Rightarrow 16 = 4x \Rightarrow x = 4$$ 14. **Step 3:** Substitute $x=4$ back: $$LM = 5(4) + 4 = 20 + 4 = 24$$ $$NP = 9(4) - 12 = 36 - 12 = 24$$ 15. **Step 4:** Use the intersecting chords theorem: product of segments of one chord equals product of segments of the other. If $CN = 13$, and $CY$ is unknown, then: $$LM \times CY = NP \times CN$$ Assuming $LM$ and $NP$ are full chord lengths, and $CY$ is segment of $LM$ corresponding to $CN$ segment of $NP$. 16. **Step 5:** Without additional segment lengths, assume $CY = CN = 13$ (if $CY$ is equal to $CN$ by symmetry or problem context). **Final answers:** - Problem 1: None of the options match $x \approx 20.6$; recheck problem context. - Problem 2: Statements i and ii are true. - Problem 3: $x=4$, $LM=NP=24$, $CY$ depends on further info; if $CY=13$, then consistent.