1. **Problem 1:** A line from the center of a circle cuts a chord into two parts of lengths 3 cm and 2 cm. The perpendicular distance from the center to the chord is 1 cm. Find the radius of the circle. 2. **Step 1:** Let the radius be $r$. The chord is divided into two parts: 3 cm and 2 cm, so the half-length of the chord is $\frac{3+2}{2} = 2.5$ cm. 3. **Step 2:** The perpendicular from the center to the chord forms a right triangle with half the chord and the radius. Using the Pythagorean theorem: $$r^2 = 1^2 + 2.5^2 = 1 + 6.25 = 7.25$$ 4. **Step 3:** Therefore, the radius is: $$r = \sqrt{7.25} \approx 2.69 \text{ cm}$$ 5. **Problem 2:** A line from the center meets a chord, dividing it into two parts. The distances given are 3 cm, 2 cm, and 4.5 cm. Find the lengths of the two parts of the chord. 6. **Step 1:** The 4.5 cm is the distance from the center to the chord (perpendicular). The 3 cm and 2 cm are segments along the chord from the foot of the perpendicular to the ends. 7. **Step 2:** Let the two parts of the chord be $x$ and $y$. Since the perpendicular divides the chord into two parts, and the segments from the foot to the ends are 3 cm and 2 cm, the chord length is $x + y = 3 + 2 = 5$ cm. 8. **Step 3:** The radius $r$ can be found using the Pythagorean theorem on the triangle formed by the radius, the perpendicular, and half the chord: $$r^2 = 4.5^2 + (\frac{5}{2})^2 = 20.25 + 6.25 = 26.5$$ 9. **Step 4:** The radius is: $$r = \sqrt{26.5} \approx 5.15 \text{ cm}$$ 10. **Problem 3:** AB is a diameter of the circle extended to point P. The tangent from P touches the circle at Q. Find the length of the tangent segment PQ if AP = 4.5 cm and the radius is $r$. 11. **Step 1:** Since AB is diameter, $AB = 2r$. Given $AP = 4.5$ cm, and $P$ lies on the extension of $AB$ beyond $B$. 12. **Step 2:** The length of tangent from an external point is given by: $$PQ = \sqrt{PA^2 - r^2}$$ 13. **Step 3:** Using $r = 2.69$ cm (from problem 1) or $r = 5.15$ cm (from problem 2), calculate $PQ$: - Using $r=2.69$ cm: $$PQ = \sqrt{4.5^2 - 2.69^2} = \sqrt{20.25 - 7.24} = \sqrt{13.01} \approx 3.61 \text{ cm}$$ - Using $r=5.15$ cm: $$PQ = \sqrt{4.5^2 - 5.15^2} = \sqrt{20.25 - 26.52}$$ which is not possible (negative under root), so radius from problem 1 is used. 14. **Final answers:** - Radius of circle (Problem 1): $\approx 2.69$ cm - Lengths of chord parts (Problem 2): 3 cm and 2 cm (given), radius $\approx 5.15$ cm - Length of tangent $PQ$ (Problem 3): $\approx 3.61$ cm