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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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