Question

If one diagonal of a trapezium divides the diagonal in the ratio 1:3. Prove that one diagonal is thrice the other.

Answer 1

Given that the diagonal BD the diagonal AC in AO : OC in 3 : 1
To prove : AB = 3CD
Proof : in Δ AOC and Δ DOC
Δ AOB = Δ COD
Δ OBA = Δ ODC (Because DC is parallel to AB, and DB is transversal so these are alternates)
Therefore,
Δ AOB is similar to COD (By AA similarity)
Now, AO/OC = AB/DC (Because in similar triangles sides are proportional)
3/1 = AB/DC (Given that AO : OC = 3 : 1)
So, AB = 3DC
Hence proved.

Answer 2

Given that the diagonal BD the diagonal AC in AO : OC in 3 : 1

To prove : AB = 3CD

Proof : in Δ AOC and Δ DOC

Δ AOB = Δ COD

Δ OBA = Δ ODC (Because DC is parallel to AB, and DB is transversal so these are alternates)

Therefore,

Δ AOB is similar to COD (By AA similarity)

Now, AO/OC = AB/DC (Because in similar triangles sides are proportional)

3/1 = AB/DC (Given that AO : OC = 3 : 1)

So, AB = 3DC  

Hence proved.

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