SECTION – A (1 mark × 10 = 10 marks)
Choose the correct answer.
1. In ΔABC, D is midpoint of AB. If DE ∥ BC, then E lies on:
(a) AC (b) AB (c) BC (d) AD
2. Which statement shows the Midpoint Theorem?
(a) If a line divides two sides proportionally, it is parallel.
(b) A line joining midpoints of two sides of a triangle is parallel to the third side.
(c) Exterior angle = sum of interior angles.
(d) Diagonals bisect each other.
3. If DE ∥ BC in ΔABC and AD = 5 cm, DB = 5 cm, then D is:
(a) Foot of perpendicular (b) Midpoint of AB
(c) Vertex (d) Angle bisector
4. A line joins midpoints of two sides of a triangle. This line equals:
(a) Twice third side (b) Half third side
(c) Same as third side (d) Opposite side
5. In ΔABC, D and E are midpoints of AB and AC. DE equals:
(a) BC (b) ½ BC (c) 2 BC (d) AB
6. If DE ∥ BC and AD/DB = AE/EC, the theorem used is:
(a) Pythagoras
(b) Midpoint Theorem
(c) Converse Midpoint Theorem
(d) Triangular Inequality
7. In ΔPQR, M is midpoint of PQ and MN ∥ QR. N lies on:
(a) PR (b) QR (c) PQ (d) PR extended
8. If DE = ½ BC, D and E must be:
(a) Vertices (b) Midpoints
(c) Foot of perpendiculars (d) Exterior points
9. Converse of Midpoint Theorem proves:
(a) Two angles are equal
(b) A line is perpendicular
(c) A line is parallel to third side and divides two sides proportionally
(d) All sides are equal
10. If D and E are midpoints of two sides of a triangle, the triangle formed by DE and third side is:
(a) Isosceles (b) Parallel
(c) Similar (d) Congruent
SECTION – B (2 marks × 5 = 10 marks)
11. State the Midpoint Theorem and draw a neat labelled diagram.
12. State the Converse of the Midpoint Theorem with a diagram.
13. In ΔABC, D and E are midpoints of AB and AC. If BC = 12 cm, find DE.
14. In ΔXYZ, XY = 8 cm. M and N are midpoints of XY and XZ. If YZ = 14 cm, find MN.
15. If DE ∥ BC in ΔABC and AD = 9 cm, DB = 9 cm, AE = 7 cm, find EC.
SECTION – C (3 marks × 5 = 15 marks)
16. In ΔABC, D is midpoint of AB. A line through D parallel to BC meets AC at E.
Prove that AE/EC = AD/DB.
17. In ΔPQR, M and N are midpoints of PQ and PR.
Prove that MN = ½ QR.
18. In ΔXYZ, a line parallel to YZ meets XY at A and XZ at B.
Given XA/AY = 1, prove XB = BY (using converse).
19. In ΔABC, AB = 10 cm and AC = 16 cm. D and E are midpoints of AB and AC.
Find DE and prove DE ∥ BC.
20. In ΔLMN, LM = 18 cm and MN = 12 cm. P is midpoint of LM and Q lies on LN such that PQ ∥ MN.
Find LQ / QN.
SECTION – D (5 marks × 3 = 15 marks)
21. PROVE USING MIDPOINT THEOREM:
In ΔABC, D and E are midpoints of AB and AC.
A line through E meets BC at F.
Show that DF ∥ AC.
22. APPLICATION PROBLEM – HEIGHT OF A BUILDING:
A student stands 20 m away from a building and measures the height using similar triangles.
He notices that a stick of height 1 m casts a shadow of 0.5 m.
The building casts a shadow of 12 m.
Using the Midpoint Theorem / similarity, find the height of the building.
23. PROVE THE CONVERSE:
In ΔABC, a line DE divides AB and AC in the ratio AD/DB = AE/EC.
Prove that DE ∥ BC.
