1. 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.

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. 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.

4. Mohit has drawn two straight lines from an external point of a circle, which intersect the circle at points A, B and C, D respectively. Using reasoning, prove that two angles of ∆XAC and ∆XBD are equal.

5. 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 \(BC\).

6. 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 \(BC\).

7. 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 \(OA \parallel RQ\).

8. 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.

9. 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.

10. 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.

11. 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

12. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

13. 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.

14. Here is the English translation: > Two pillars of equal height are located directly opposite each other at points A and B on either side of a 120-meter wide road. From point C, on the line joining their bases, the angles of elevation to the tops of the pillars at A and B are 60° and 30°, respectively. Find the length of AC.

15. 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 \(\angle QST\)?

16. 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

17. 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

18. 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 \(AB\).

19. Triangles \(∆ABC\) and \(∆DBC\) are drawn on the same base \(BC\) and on the same side of it. Let \(E\) be any point on side \(BC\). From point \(E\), lines parallel to \(AB\) and \(BD\) are drawn, intersecting sides \(AC\) and \(DC\) at points \(F\) and \(G\) respectively. Prove that: \(AD ∥ FG\).

20. 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: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

21. 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.

22. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?

23. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.

24. Two radii, OA and OB, of a circle centered at O form an angle of 130°. Tangents are drawn at points A and B, and they intersect at point T. Find the measures of \(\angle\)ATB and \(\angle\)ATO.

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. The two tangents drawn from an external point to a circle are unequal.

27. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

28. 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.

29. 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.

30. From two points located on the same side and along the same horizontal line passing through the base of a vertical pillar, the angles of elevation to the top of the pillar are respectively \(\theta\) and \(\phi\). If the height of the pillar is \(h\), find the distance between the two points.