1. A straight line that divides two sides of a triangle in the same ratio is parallel to the third side.
(a) perpendicular (b) equal (c) diagonal (d) parallel
2. If a straight line divides two sides of a triangle in the same ratio, then that line will be _____ to the third side of the triangle.
3. Prove that the line segment joining the midpoints of two sides of a triangle is equal to half of the third side.
4. "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]"
5. 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
6. 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?
7. 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.
8. 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.
9. A straight line parallel to any side of a triangle divides the other two sides or their extensions --.
10. 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 \(AD\).
11. A straight line parallel to any side of a triangle divides the other two sides or their extensions________.
12. If two triangles have their bases on the same straight line and share the same vertex (the opposite vertex), then the ratio of their areas is equal to the ratio of the lengths of their bases.
13. 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
14. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).
(a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above
15. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP = 18 \) cm, \( QC = 9 \) cm, and \( AQ = 2 \times PB \), find the length of \( PB \).
(a) 6 cm (b) 12 cm (c) 18 cm (d) 9 cm
16. 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
17. Draw a median \(AD\) of \(\triangle ABC\). If a straight line parallel to \(BC\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that \(AD\) bisects the line segment \(PQ\).
18. A right circular cone is divided into three parts by two planes drawn parallel to its base, which trisect its height. Show that the ratio of the volumes of the three resulting parts is 1:7:19.
19. 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
20. Prove that, in a trapezium, the straight line joining the midpoints of the diagonals is parallel to the parallel sides.
21. 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: \(AD ∥ FG\).
22. A straight line drawn parallel to the parallel sides of a trapezium divides the other two (non-parallel) sides _________.
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. Translate: A segment drawn parallel to any side of a triangle divides the other two sides (or their extensions) in _____ ratio.
25. In triangle ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. AP = QC, AB = 12 cm, AQ = 2 cm. Find the length of CQ.
(a) 4 cm (b) 6 cm (c) 9 cm (d) None of the above
26. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?
(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17
27. 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
28. In triangle ∆ABC, a straight line parallel to side BC intersects AB and AC at points P and Q respectively. Given that \(\frac{AQ}{QC} = \frac{3}{4}\) and AB = 21 cm, find the length of PB.
29. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points X and Y respectively. If AX = 2.4 cm, AY = 3.2 cm, and YC = 4.8 cm, find the length of AB.
30. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)
