9. ABCD is a trapezium in which AB || DC and its
diagonals intersect each other at the point O. Show
AO _co
that BO
DO
Answers
Answer:Given: □ABCD is a trapezium where, AB ll CD
Diagonals AC and BD intersect at point O.
Construction: Draw a line EF passing through O and also parallel to AB.
Now, AB ll CD, since by construction, EF ll AB ⇒ EF ll CD
Consider the ΔADC,
EO ll DC
Thus, by Basic proportionality theorem, (AE / ED) = (AO / OC) .... (i)
Now, consider Δ ABD,
EO ll AB,
Thus, by Basic proportionality theorem, (AE / ED) = (BO / OD) .... (ii)
From (i) and (ii), we have, (AO / OC) = (BO / OD) (since L.H.S of i and ii are equal)
Hence we proved that, (AO / OC) = (BO / OD)
Question :
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O.Show AO/BO = CO/DO
Given :
ABCD is a trapezium where AB||DC and diagonals AC and BD intersect each other at O.
To prove :
Solution :
From the point O,draw a line XO touching AD at X,in such a way that,XO||DC||AB
In triangle ADC,we have OX||DC
Therefore, by using basic proportionality theorem
--(i)
Now,in triangle ABD OX||AB
By using basic proportionality theorem
--(ii)
From equation (i) and (ii), we get,
Hence Proved.
Additional Information :
Basic proportionality theorem :
If a line is drawn parallel to one side of the triangle , Then the other sides are divided in the same ratio.
Here, We prove that
In trapezium ABCD , AO/BO = CO/DO
Using the Basic proportionality theorem.
We constructed OX || AB and proceeded with the problem.
Check out the attachment for detailed explanation.