1. More than two tangents can be drawn to a circle that are parallel to a given straight line.
2. Prove that two tangents can be drawn from any external point to a given circle.
3. Three tangents parallel to a given straight line can be drawn to a circle.
4. 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.
5. 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 \(OA \parallel RQ\).
6. If two circles neither intersect nor touch each other, the maximum number of common tangents that can be drawn to them is _________.
7. 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.
8. 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 \(\angle QST\)?
9. 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.
10. 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.
11. If two circles do not intersect or touch each other, then a maximum of _____ common tangents can be drawn between them.
12. The centers of two circles are P and Q. The circles intersect at points A and B. Through point A, two straight lines parallel to the line segment PQ intersect the circles at points C and D. Prove that CD = 2PQ.
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 line segment \(BC\).
14. 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 \(AB\).
15. If two circles neither intersect nor touch each other, a maximum of 4 common tangents can be drawn.
16. The maximum number of tangents that can be drawn to a circle from an external point is _____.
17. 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.
18. The centers of two circles are P and Q; the circles intersect at points A and B. A line parallel to the line segment PQ is drawn through point A, and it intersects the two circles at points C and D. Prove that CD = 2PQ.
19. 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.
20. 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 \(BC\).
21. "P and Q are two points on a straight line. From points P and Q, perpendiculars PR and QS are drawn respectively. PS and QR intersect at point O. OT is drawn perpendicular to PQ. Prove that: \[ \frac{1}{OT}=\frac{1}{PR}+\frac{1}{QS} \]"
22. 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: - \(OD = OC = OE\) - \(x - y = 4^\circ\)
23. I have drawn two circles with centers A and B that externally touch each other at point O. A straight line is drawn through point O, which intersects the two circles at points P and Q respectively. Prove that AP is parallel to BQ.
24. 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.
25. I draw a circle with center O where two radii, OA and OB, are perpendicular to each other. Tangents are drawn at points A and B, which intersect each other at point T. Prove that AB = OT and that they bisect each other perpendicularly.
26. I have drawn two circles that intersect each other at points G and H. Then I drew a straight line through point G which intersects the two circles at points P and Q. Next, I drew another straight line through point H, parallel to PQ, which intersects the two circles at points R and S. Prove that PQ = RS.
27. 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.
28. 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
29. 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
30. 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
