1. From any external point of a circle, a maximum of _____ tangents can be drawn to the circle.
2. 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.
3. The angle formed between the two tangents drawn from an external point to a circle is bisected by the straight line segment connecting that point to the center of the circle.
4. Prove that the two tangents drawn from an external point to a circle are equal in length from the point to the points of contact on the circle.
5. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.
6. From an external point, a maximum of _____ tangents can be drawn to a circle.
7. The maximum number of tangents that can be drawn to a circle from an external point is _____.
8. More than two tangents can be drawn to a circle that are parallel to a given straight line.
9. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the line segment \(BC\).
10. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\). Given \(PA = 9\) cm and \(\angle APB = 60°\), find the length of \(AB\).
11. Prove that from an external point to a circle, the two tangents drawn are equal in length, and the line segments connecting the external point to the points of contact form equal angles at the center.
12. Only one tangent can be drawn to a circle from any external point.
13. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the chord \(BC\).
14. 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.
15. From an external point \(A\), two tangents \(AP\) and \(AQ\) are drawn to a circle centered at \(O\), touching the circle at points \(P\) and \(Q\) respectively. If \(PR\) is a diameter of the circle, prove that \(OA \parallel RQ\).
16. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)
17. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.
18. More than two tangents can be drawn to a circle that are parallel to a given straight line.
19. In triangle ABC, perpendiculars BE and CF are drawn respectively to sides AC and AB. Prove that the four points B, C, E, and F lie on a circle. From this, also prove that triangles ∆AEF and ∆ABC have two equal angles each.
20. 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.
21. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?
22. No of tangent can be drawn to a circle from an external point-
(a) 1 (b) 3 (c) 4 (d) 2
23. 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
24. 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?
25. From an external point P, two tangents PA and PB are drawn to a circle centered at O. If PA = 9 cm and ∠APB = 60°, then what are the measures of ∠PAB and the length of chord AB?
(a) 9 cm (b) 3 cm (c) 6 cm (d) 12 cm
26. The two tangents drawn from an external point to a circle are unequal.
27. No tangent can be drawn to a circle from an external point.
28. In the given figure, the angle between the two radii OA and OB of a circle centered at O is 130°. Tangents drawn at points A and B intersect at point T. Calculate and write the measures of \(\angle\)ATB and \(\angle\)ATO.
29. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).
30. A semicircle is drawn with AB as the diameter. From any point C on AB, a perpendicular is drawn to AB, which intersects the semicircle at point D. Prove that CD is the mean proportional between AC and CB.
