1. Let's draw a circle with a radius of 3.2 cm. Then, draw a tangent to the circle at any point on its circumference.
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. 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.
4. 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.
5. 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
6. 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.
7. 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?
8. Draw a circle with a radius of 4 cm. From a point located 6.5 cm away from the center of the circle, draw two tangents to the circle.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.
15. 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
16. 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.
17. 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.
18. If the chord of the larger circle of two concentric circles with radii 3 cm and 5 cm is a tangent to the smaller circle, what is its length?
(a) 4 cm (b) 6 cm (c) 8 cm (d) 12 cm
19. The distance from the center of a circle to an external point is 13 cm. The length of the tangent from that point to the circle is 12 cm. The radius of the circle is—
(a) 25 cm (b) 1 cm (c) 5 cm (d) \(\cfrac{13}{12}\) cm
20. 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
21. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.
22. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.
23. 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?
24. 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.
25. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?
26. Prove:The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact.
27. If a circle has a radius of 5 cm and a tangent is drawn from an external point \(P\) to the circle with a length of 12 cm, what is the distance from the center to point \(P\)?
28. Construct a right-angled triangle in which the two arms adjacent to the right angle are 7 cm and 9 cm. Draw an incircle (inscribed circle) of that triangle and measure its radius. (Each construction step must be marked.)
29. 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.
30. AB = 5 cm, ∠BAC = 30°, ∠ABC = 60°; AB = 5 cm, ∠BAD = 45°, ∠ABD = 45°; Draw ∆ABC and ∆ABD in such a way that points C and D lie on opposite sides of AB. Draw the circumcircle of ∆ABC and write the position of point D with respect to that circle. Also, observe and describe any other noteworthy properties.
