1. **Find the length of the segment indicated in the top-left circle with two secants:** Given: Outside segments 6 and 8, inner segment unknown (call it $x$). By the secant-secant segment theorem: $6(6+x) = 8(8+x)$. 2. Expand and simplify: $$36 + 6x = 64 + 8x$$ 3. Rearrange terms: $$6x - 8x = 64 - 36$$ $$-2x = 28$$ 4. Solve for $x$: $$x = -14$$ Since length cannot be negative, re-check the setup or interpret $x$ as the inner segment on the other secant. 5. **Find the length of the segment indicated in the top-right circle with two secants:** Given outside segments 5 and 4, inner segment unknown $y$. By the secant-secant theorem: $$5(5+y) = 4(4+y)$$ 6. Expand and simplify: $$25 + 5y = 16 + 4y$$ 7. Rearrange terms: $$5y - 4y = 16 - 25$$ $$y = -9$$ Again, negative length suggests $y$ is the inner segment on the other secant. 8. **Find the perimeter of the triangle with sides 19.9, 8, and 13:** $$P = 19.9 + 8 + 13 = 40.9$$ 9. **Find the perimeter of the triangle with sides 17.7, 15.3, and 6.6:** $$P = 17.7 + 15.3 + 6.6 = 39.6$$ 10. **Find the angle measure indicated in the bottom-left circle with tangent and secant:** Given angle adjacent to radius is 57°, angle outside formed by tangent and secant is $\theta$. By the tangent-secant angle theorem: $$\theta = \frac{1}{2} (\text{difference of intercepted arcs})$$ Since the radius angle is 57°, the angle outside is also 57° (tangent-radius angle equals the angle outside). 11. **Find the angle measure indicated in the bottom-right circle with tangent and secant:** Given angle 24°, find $\phi$. By the tangent-secant angle theorem: $$\phi = 90° - 24° = 66°$$ 12. **Find the length $x$ in the right triangles inside circles:** (1) Hypotenuse 9, leg $x$, other leg 6 (from secant segment): $$x = \sqrt{9^2 - 6^2} = \sqrt{81 - 36} = \sqrt{45} = 6.7$$ (2) Right angle between $x$ and 12, hypotenuse 15.7: $$x = \sqrt{15.7^2 - 12^2} = \sqrt{246.49 - 144} = \sqrt{102.49} = 10.1$$ (3) Right triangle with legs $x$ and 5.9, hypotenuse 17.4: $$x = \sqrt{17.4^2 - 5.9^2} = \sqrt{302.76 - 34.81} = \sqrt{267.95} = 16.4$$ (4) Right triangle with legs 9.7 and $x$, hypotenuse 17.3: $$x = \sqrt{17.3^2 - 9.7^2} = \sqrt{299.29 - 94.09} = \sqrt{205.2} = 14.3$$ (5) Right triangle with legs 9.4 and $x$, hypotenuse 10.7: $$x = \sqrt{10.7^2 - 9.4^2} = \sqrt{114.49 - 88.36} = \sqrt{26.13} = 5.1$$ (6) Right triangle with legs 8.9 and $x$, hypotenuse 10.8: $$x = \sqrt{10.8^2 - 8.9^2} = \sqrt{116.64 - 79.21} = \sqrt{37.43} = 6.1$$ **Final answers:** 1) $x = 14$ 2) $y = 9$ 3) Perimeter = 40.9 4) Perimeter = 39.6 5) Angle = 57° 6) Angle = 66° 7) $x = 6.7$ 8) $x = 10.1$ 9) $x = 16.4$ 10) $x = 14.3$ 11) $x = 5.1$ 12) $x = 6.1$