1. "Prove that, in a triangle, a straight line drawn parallel to the second side through the midpoint of one side bisects the third side. [Prove using Thales's Theorem]"

2. 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

3. 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.

4. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).

5. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

6. Two circles with centers X and Y intersect at points A and B. The midpoint S of the line segment XY is joined with point A, and from point A, a perpendicular is drawn on line SA, which intersects the two circles at points P and Q. Prove that PA = AQ.

7. 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.

8. A line parallel to side \(BC\) of triangle \(∆ABC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. The length of \(PB\) is twice the length of \(AP\), and the length of \(QC\) is 3 units more than the length of \(AQ\). Find the length of side \(AC\).

9. Prove that, in a trapezium, the straight line joining the midpoints of the diagonals is parallel to the parallel sides.

10. 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

11. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

12. 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?

13. Prove that the line segment joining the midpoints of two sides of a triangle is equal to half of the third side.

14. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

15. A line parallel to side BC of triangle ABC intersects sides AB and AC at points D and E, respectively. If AD is half of AE, then the ratio BD:EC will be –

(a) 1:2 (b) 2:1 (c) 2:3 (d) 1:3

16. Rajat has drawn a straight line segment PQ whose midpoint is R. He then drew two circles using PR and PQ as diameters. I drew a straight line passing through point P that intersects the first circle at point S and the second circle at point T. Prove with reasoning that PS = ST.

17. ABC is an acute-angled triangle. AP is the diameter of the circumcircle of triangle ABC. BE and CF are perpendiculars dropped to sides AC and AB respectively, and they intersect at point Q. Prove that BPCQ is a parallelogram.

18. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.

19. 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.

20. On a circle with center O, A is a point on the circle. A tangent is drawn at point A, and point X is any point on that tangent. A secant drawn from point X intersects the circle at points Y and Z. If P is the midpoint of segment YZ, prove that XAPO (or XAOP) is a cyclic quadrilateral.

21. I have drawn a cyclic quadrilateral ABCD, and extended its side BC up to point E. Prove that the bisectors of ∠BAD and ∠DCE meet on the circle.

22. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

23. 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

24. If E and F are the midpoints of sides AB and AD respectively in parallelogram ABCD, and the area of parallelogram ABCD is 1600 square centimeters, then what is the area of triangle AEF?

(a) 400 sq cm (b) 200 sq cm (c) 300 sq cm (d) None of the above

25. 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

26. ABCD is a cyclic quadrilateral. The side BA is extended up to point F. AE is drawn parallel to CD, and ∠ABC = 92°, ∠FAE = 20°. Find the measure of ∠BCD.

27. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

28. 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.

29. 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?

30. In the parallelogram ABCD, point E is the midpoint of side BC. The line DE meets the extended line AB at point F. The length of AF is equal.

(a) \(\cfrac{3}{2}\)AB (b) 2AB (c) 3AB (d) \(\cfrac{5}{4}\)AB