In Fig.7.2, two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the midpoint of both the line-segments AB and CD
Solution:
Given, AB and CD are two lines which intersect each other at the point O such that BC || DA.
BC = DA
We have to show that O is the midpoint of both the line segments AB and CD.
Considering triangles OBC and OAD,
Given, BC = DA
We know that if two lines are parallel, then alternate opposite angles are equal.
∠CBO = ∠DAO
∠BCO = ∠ADO
ASA criterion states that two triangles are congruent, if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.
By ASA criterion, ∆ OBC ≅ ∆ OAD
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCT,
OB = OA
OC = OD
This implies AB = OB + OA and CD = OD + OC
Therefore, O is the midpoint of AB and CD
✦ Try This: In figure The sides BA and CA have been produced such that BA=AD and CA=AE prove that segment DE∥BC
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.3 Sample Problem 2
In Fig.7.2, two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the midpoint of both the line-segments AB and CD
Summary:
In Fig.7.2, two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. It is shown that O is the midpoint of both the line-segments AB and CD
☛ Related Questions:
- In Fig.7.3, PQ > PR and QS and RS are the bisectors of ∠Q and ∠R, respectively. Show that SQ > SR
- ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE
- In Fig.7.4, D and E are points on side BC of a ∆ ABC such that BD = CE and AD = AE. Show that ∆ ABD . . . .