1. Problem 1: Two chords intersect inside the circle where one chord is split into lengths 6 and 3 and the other chord is split into lengths 4 and x. 2. By the intersecting chords theorem we have $6\cdot 3=4\cdot x$. 3. Compute $18=4x$. 4. Solve $x=\dfrac{18}{4}=\dfrac{9}{2}=4.5$. Answer: $x=4.5$. 1. Problem 2: Two secants are drawn from the same external point with one secant giving a known external segment 12 and a known interior segment 6 so that its whole length is $12+6=18$, and the other secant has external segment $x$ and the same nearer interior segment 6 so its whole length is $x+6$. 2. By the secant-secant power theorem we have $12\cdot 18 = x\cdot (x+6)$. 3. Compute $216 = x^2+6x$. 4. Rearranged this gives $x^2+6x-216=0$. 5. Solve with the quadratic formula $x=\dfrac{-6\pm\sqrt{6^2-4(1)(-216)}}{2}$. 6. Compute the discriminant $\sqrt{900}=30$. 7. The positive root is $x=\dfrac{-6+30}{2}=\dfrac{24}{2}=12$ and we discard the negative root $x=-18$ because lengths are positive. Answer: $x=12$. 1. Problem 3: Two chords intersect inside the circle so that one chord is split into lengths 16 and x and the other chord is split into lengths 8 and 10. 2. By the intersecting chords theorem we have $16\cdot x = 8\cdot 10$. 3. Compute $16x=80$. 4. Solve $x=\dfrac{80}{16}=5$. Answer: $x=5$. 1. Problem 6: Two chords intersect inside with segments 12 and x on one chord and 8 and 7 on the other. 2. By the intersecting chords theorem $12\cdot x = 8\cdot 7$. 3. Compute $12x=56$. 4. Solve $x=\dfrac{56}{12}=\dfrac{14}{3}\approx 4.6667$. Answer: $x=\dfrac{14}{3}\approx 4.6667$. 1. Problem 7: Two intersecting chords have segments 4 and x on one chord and 5 and 6 on the other. 2. By the intersecting chords theorem $4\cdot x = 5\cdot 6$. 3. Compute $4x=30$. 4. Solve $x=\dfrac{30}{4}=7.5$. Answer: $x=7.5$. 1. Problem 8: From external point O a secant goes through A to R with external piece 9 and internal piece 16 so the whole secant length is $9+16=25$, and the other line from O is a tangent to the circle with length x. 2. By the power of a point for a tangent and secant we have $x^2 = 9\cdot 25$. 3. Compute $x^2=225$. 4. Solve $x=\sqrt{225}=15$. Answer: $x=15$.