1. Prove that if a perpendicular is drawn from the center of a circle to a chord that is not a diameter, then the perpendicular bisects the chord.

2. If a perpendicular is drawn from the center of a circle to a chord that is not a diameter, it bisects the chord.

3. Prove that if a line drawn from the center of a circle bisects a chord (which is not the diameter), then the line is perpendicular to the chord.

4. Prove that if a chord (which is not a diameter) of a circle is bisected by a straight line drawn from the center of the circle, then that line is perpendicular to the chord.

5. Prove that if a straight line passing through the center of a circle bisects a chord that is not a diameter, then the line is perpendicular to the chord.

6. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

7. 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 \(\angle QST\)?

8. AB is a chord of a circle with center O. From point O, a perpendicular OP is drawn to the chord AB. The extended line OP intersects the circle at point C. If AB = 6 cm and PC = 1 cm, then what is the radius of the circle?

9. The radius of a circle is 13 cm. If the length of a chord AB in that circle is 10 cm, find the perpendicular distance from the center of the circle to the chord.

10. If the perpendicular drawn from the center of a circle with a radius of 10 cm to a chord is 6 cm long, then the length of the chord is ______.

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. 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

13. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

14. AB is a chord (not a diameter) of a circle with center O. If the length of chord AB is 8 cm and the radius of the circle is 5 cm, find the perpendicular distance from the center O to the chord AB.

15. If the radius of a circle is \(\sqrt{2}a\) and the perpendicular distance from the center to a chord is \(a\), then determine the length of the chord.

16. Prove that if a perpendicular is drawn from the right-angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of the perpendicular are similar to each other.

17. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

18. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

19. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.

20. In a circle, the chords AB and CD are perpendicular to each other. From their point of intersection P, a perpendicular is drawn to AD and extended; it intersects BC at point E. Prove that E is the midpoint of BC.

21. 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 \(BC\).

22. 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.

23. A circle with center O has a radius of 10 cm. PQ is a chord with a length of 16 cm. The length of the perpendicular drawn from O to PQ is —

(a) 8 cm (b) 10 cm (c) 16 cm (d) 6 cm

24. A chord is drawn in a circle with a radius of 5 cm, at a distance of 3 cm from the center. What will be the length of that chord?

(a) 8 cm (b) 2 cm (c) 5 cm (d) 3 cm

25. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

26. If a perpendicular is drawn from the point of intersection of two ogives to the x-axis, then the foot of that perpendicular on the x-axis represents the median.

27. AB and CD are two chords of a circle with center O. OM and ON are perpendiculars drawn from the center to the chords such that OM = ON. Prove that AB = CD.

28. 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?

29. 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 \(BC\).

30. If the radius of a circle is \(r\sqrt{2}\) and the perpendicular distance from the center to a chord is \(r\), then what is the length of the chord?