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In figure ABC and DBC are two triangles...

In figure ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that `(a r(A B C))/(a r(D B C))=(A O)/(D O)`.

Text Solution

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Draw `AL bot BC and MC bot BC`
In `triangleOLA and triangleOMD`
` angleALO = angle= 90^(@)`
and ` angle AOL = angleDOM` ( vertically opposite angle)
`angleOLA ~ angleOMD` ( AAA similarity criterion)
`(AL)/(DM) = (AO)/(DO)` …(1)
Now, ` (ar(triangleABC))/(ar(triangleDBC))= (1/2xx(BC)xx(AL))/(1/2xx(BC)xx(DM))(AL)/(DM)= (AO)/(DO)`
Hence , ` (ar(triangleABC)/(ar(triangleDBC))= (AO)/(DO)` Hence proved.
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