Ex 6.2, 10
The diagonals of a quadrilateral ABCD intersect each other at the point O such that π΄π/π΅π = πΆπ/π·π . Show that ABCD is a trapezium
Given: ABCD is a quadrilateral
where diagonals AC & BD intersect at O
& π΄π/π΅π=πΆπ/π·π
To prove: ABCD is a trapezium
Construction: Let us draw a line EF II AB passing through point O.
Proof: Given π΄π/π΅π=πΆπ/π·π
β π΄π/πΆπ=π΅π/π·π
Now,
in β π΄π·π΅
EO II AB
π΄πΈ/π·πΈ=π΅π/π·π
β π΄πΈ/π·πΈ=π΄π/πΆπ
Thus in Ξ ADC,
Line EO divides the triangle in the same ratio
β΄ EO II DC
Now, EO II DC
But, we know that EO II AB
β EO II AB II DC
β AB II DC
Hence,
one pair of opposite sides of quadrilateral ABCD are parallel
Therefore ABCD is a trapezium .
Hence proved

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.