01 · TRAPEZIUM PROOF
Diagonals of a Trapezium
A classic NCERT problem involves proving the ratio of segments formed by intersecting diagonals in a trapezium:
PROOF 1
ABCD is a trapezium with AB ∥ DC. Diagonals AC and BD intersect at O. Prove AO / OC = BO / OD.
Proof using Similarity:
Consider △AOB and △COD:
1. ∠AOB = ∠COD (Vertically opposite angles)
2. ∠OAB = ∠OCD (Alternate interior angles since AB ∥ DC)
By AA similarity criterion, △AOB ~ △COD.
Therefore, ratio of corresponding sides is equal:
AO / CO = BO / DO ⟹ AO / OC = BO / OD (Proved)
Therefore, ratio of corresponding sides is equal:
AO / CO = BO / DO ⟹ AO / OC = BO / OD (Proved)
02 · NCERT EXERCISES
Step-by-Step Geometry Problems
EXAMPLE 2
In △ABC, DE ∥ BC with AD = 1.5 cm, DB = 3 cm, AE = 1 cm. Find EC.
Since DE ∥ BC, by Basic Proportionality Theorem (BPT):
AD / DB = AE / EC
1.5 / 3 = 1 / EC
1 / 2 = 1 / EC ⟹ EC = 2 cm
1.5 / 3 = 1 / EC
1 / 2 = 1 / EC ⟹ EC = 2 cm
03 · QUICK SUMMARY
Key Takeaways
01
In trapeziums (AB ∥ DC), triangles formed by intersecting diagonals are similar (△AOB ~ △COD).
02
Whenever a line is parallel to one side of a triangle, immediately apply BPT (AD/DB = AE/EC).
03
Always write down Given, To Prove, Construction, and Proof clearly in exam answers!