1. Draw a circle with a radius of 4 cm. From a point located 9 cm away from the center of the circle, draw a tangent to the circle.
2. The radius of a circle with center \(O\) is 5 cm. Point \(P\) is located at a distance of 13 cm from \(O\). From point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle. Find the area of the quadrilateral \(PQOR\).
(a) \(60\) square cm (b) \(30\) square cm (c) \(120\) square cm (d) \(150\) square cm
3. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.
4. The radius of a circle centered at point O is 5 cm. Point P is located 13 cm away from point O. PQ and PR are two tangents drawn from point P to the circle. What is the area of quadrilateral PQOR?
5. Let’s draw a circle with a radius of 2.8 cm. Then, take a point that is 7.5 cm away from the center of the circle. From that external point, draw two tangents to the circle.
6. The radius of a circle with center O is 5 cm. From an external point P, which is located at a certain distance from point O, two tangents PQ and PR are drawn to the circle. The quadrilateral PQOR has an area of 60 square centimeters. Find the distance from point O to point P.
7. I have drawn a circle with center O and radius 6 cm. From a point P located 10 cm away from the center O, a tangent PT is drawn to the circle. Calculate and write the length of the tangent PT.
8. Let's draw a circle with a radius of 2.5 cm. Take a point outside the circle that is 6.5 cm away from the center. Then, draw a tangent to the circle from that external point and measure the length of the tangent using a ruler.
9. The radius of a circle centered at point O is 5 cm. Point P lies at a distance of 13 cm from point O. From point P, two tangents PQ and PR are drawn to the circle. Find the area of quadrilateral PQOR.
(a) 60 square cm (b) 30 square cm (c) 120 square cm (d) 150 square cm
10. What is the length of the tangent drawn from an external point located 17 cm away from the center of a circle with a radius of 4 cm?
11. Find the length of the tangent drawn from a point located 13 cm away from the center of a circle with a radius of 5 cm.
12. Draw a circle with a radius of 2.6 cm and, from a point 6 cm away from the center (outside the circle), draw a tangent to the circle.
13. If I draw a circle centered at point O, and from a point P located 26 cm away from the center a tangent is drawn to the circle which measures 10 cm in length, then calculate and write the length of the radius of the circle.
14. Draw a straight line segment XY with a length of 4 cm, and use XY as the diameter to draw a circle. At points X and Y, draw tangents to the circle, and write about the relationship between these two tangents.
15. From a point outside a circle, a tangent of length 8 cm is drawn to the circle. If the radius of the circle is 6 cm, what is the distance from the center of the circle to the external point?
(a) 10 cm (b) 8 cm (c) 6 cm (d) 14 cm
16. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?
(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm
17. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?
(a) 145° (b) 55° (c) 110° (d) None of the above
18. From an external point located 17 cm away from the center of a circle with a diameter of 16 cm, what is the length of the tangent drawn to the circle?
19. A circle is centered at point O with a radius of 10 cm. A perpendicular is drawn from O to a chord AB, and the length of this perpendicular is 6 cm. What is the length of the chord AB?
20. Point P is an external point to a circle with center O. The distance from point P to the center of the circle is 26 cm, and the length of the tangent drawn from point P to the circle is 10 cm. The radius of the circle is ____ cm.
21. A circle is centered at O with a radius of 6 cm. From a point P located 10 cm away from the center O, determine the length of the tangent PT, where T is the point of tangency.
22. Find the length of the tangent drawn from an external point located 17 cm away from the center of a circle with a diameter of 16 cm.
23. What is the length of the tangent drawn from an external point located 17 cm away from the center of a circle with a radius of 16 cm?
24. Niyamat has drawn a circle with a radius of 13 cm. I have drawn a chord AB of length 10 cm in this circle. Calculate and write the distance from the center of the circle to this chord AB.
25. A right-angled triangle where the two sides adjacent to the right angle are 4 cm and 4 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]
26. Draw a rectangle PQRS where PQ = 4 cm and QR = 6 cm. Draw the two diagonals of the rectangle. Without drawing, calculate and write the position of the circumcenter of ∆PQR and the length of its circumradius. Then, draw the circumcircle of ∆PQR to verify.
27. From a point outside a circle, two tangents can be drawn. The line segments joining the external point to the points of contact of the tangents are equal in length, and they subtend equal angles at the center of the circle.
28. Draw a straight line segment AB of radius 3 cm. With point A as the center and radius equal to AB, draw a circle. Then, draw a tangent to the circle at point B.
29. Point P lies on a circle with center O. At point P, draw a tangent to the circle, and from that tangent, mark a segment PQ equal in length to the radius of the circle. From point Q, draw another tangent QR to the circle. Using a protractor, measure the angle ∠PQR and write down its value.
30. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.
