Sum no. 10 Exercise 6.2 class 10th



Sum No.10:-
The diagonals of  a quadrilateral ABCD intersect each other at a point O such that AO/BO=CO/DO.
show that ABCD is a trapezium.
Solution:-
Given:- ABCD is a quadrilateral, whose diagonals AC and BD intesect each other at O.
such that AO/BO=CO/DO i.e., AO/CO=BO/DO......(1)
To prove :- ABCD is a trapezium.
Construction:- from O,draw OE∥AB.

Proof:- In △ADB, we have,
             OE∥AB (construction)
∴ by BPT, we have 
AE/ED=BO/DO.....(2)
from (1) and (2),we have AO/CO=AE/ED
thus in △ADC,E and O are points dividing the sides AD and AC in the same ratio.
∴ by converse of BPT, we have
{OE∥CD but OE∥AB by construction}
hence OE∥CD∥AB ⇒ AB∥CD
∴ Quadrilatral ABCD is at Trapezium.

Comments

Post a Comment

Popular posts from this blog

Basic proportionality Theorem or Thales Theorem class 10th

Image
Imp points or results used in  Basic proportionality theorem or Thales theorem:- (i) Area of Triangle = 1/2 base* height Area of Triangle= 1/2 base*height                     =1/2 BC*AD (ii) Area of obtuse triangle =  In this figure CM is height of the triangle and AB is the base Area of triangle = 1/2 base* height                    =1/2 AB* CM (iii) Two triangles on the same base (or equal bases) and between the same parallels are equal in areas. (class 9th) There are two triangle in this figure △CDA and △CDB so,  ar(CDA)= ar(CDB) Theorem 6.1 if a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio .   Proof  Given:-    triangle ABC in which a line l parallel to side BC intersects other two sides AB and AC at D and E respec...
Read more

EXERCISE 6.2 sum no.4 (class 10th )

Image
              EXERCISE 6.2 SUM 4.    In a figure, DE∥AC and DF∥ AE, prove that BE/FE=BE/EC. Solution:-  Given:-  DE∥AC  and DF∥ AE To Prove :-  BE/FE=BE/EC Proof:- In △ABC  DE∥AC  Because ( if a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio .) BE/EC=BD/DA        .......(1) and, In △AEB DF∥ AE Because  ( if a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio .) BF/FE= BD/DA            ......(2)     From (1) and (2)   BE/FE=BE/EC Hence proved 
Read more