1. An arc of a circle is a straight line.
2. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.
3. AB is the diameter of a circle with center O. A straight line drawn from point A intersects the circle at point C, and the tangent drawn from point B intersects this line at point D. Prove that: For any such straight line, the area of the rectangle formed by AC and AD is always constant.
4. "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: \[ \frac{1}{OT}=\frac{1}{PR}+\frac{1}{QS} \]"
5. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.
6. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?
(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these
7. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?
(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1
8. In triangle \( \triangle ABC \), AD is a median. A straight line parallel to side BC intersects sides AB, AD, and AC at points P, O, and Q respectively. What is the ratio \( PO : OQ \)?
(a) 1:2 (b) 2:3 (c) 1:1 (d) None of the above
9. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?
(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm
10. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD?
(a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral
11. A ship travels 10 km north from a certain point, then 10 km west. What is the ship’s direct (straight-line) distance from the starting point?
(a) \(20\) km (b) \(10\sqrt2\) km (c) \(2\sqrt{10}\) km (d) 100 km
12. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that PB = AQ, AP = 9 units, and QC = 16 units, what is the length of PB?
(a) 12 cm (b) 6 cm (c) 8 cm (d) 10 cm
13. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?
14. More than two tangents can be drawn to a circle that are parallel to a given straight line.
15. If the angles of depression from a lighthouse to two ships located along the same straight line are 60° and 30°, and the nearer ship is 150 meters away from the lighthouse, then what is the distance of the farther ship from the lighthouse?
16. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.
17. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.
18. 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.
19. Prove that a straight line which divides two sides of a triangle in the same ratio is parallel to the third side.
20. Standing in an open field, Habul sees a flying bird at an angle of elevation of 30° towards the north, and after 2 minutes, he sees it at an angle of elevation of 60° towards the south. If the bird is flying along a straight horizontal path at a height of \(50\sqrt{3}\) meters, then calculate the speed of the bird.
21. In trapezium ABCD, AB and DC are parallel. A straight line is drawn parallel to AB, which intersects AD and BC at points E and F respectively. Prove that: \[ AE : ED = BF : FC \]
22. 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.
23. In the isosceles triangle ABC, AB = AC. A straight line parallel to BC intersects AB and AC at points P and Q respectively. Prove that the quadrilateral BCQP is cyclic (i.e., its vertices lie on a circle).
24. From a lighthouse, the angles of depression to the bases of the masts of two ships located along the same straight line are 60° and 30°, respectively. If the mast of the nearer ship is 150 meters away from the lighthouse, determine how far the mast of the farther ship is from the lighthouse and also find the height of the lighthouse.
25. Prove that a cyclic parallelogram is always a rhombus.
26. If a straight line intersects a circle at two points, the line is called a **secant** of the circle.
27. If two identical circles with a radius of 4 cm intersect such that their centers lie on a straight line, what is the length of their common chord?
(a) \(4\sqrt3\) cm (b) \(5\sqrt3\) cm (c) \(6\sqrt3\) cm (d) \(7\sqrt3\)
28. A quadratic surd is always an irrational number.
29. A parallelogram inscribed in a circle is always a square.
30. A straight line intersects \( \triangle PQR \) at points \( X \) and \( Y \) on \( PQ \) and \( PR \), respectively, such that \( \frac{PX}{XQ} = \frac{PY}{YR} \). If \( \angle PXY = \angle PRQ \), then prove that \( \triangle PQR \) is an isosceles triangle.
