Question

In parallelogram ABCD, point M is the midpoint of the side BC , point O is the intersection point of the segment AM and diagonal BD . Find the ratio BO:OD.

Answer 1

Answer: BO:OD = 1:2

Step-by-step explanation:

Here, ABCD is parallelogram, point M is the midpoint of the side BC , point O is the intersection point of the segment AM and diagonal BD.

Let N is the mid point AB ⇒ AN=NB or 2.BN=DC.  ( because AB=DC)

And join Points N and C.   (by construction)

Then, In triangles NOB and DOC,

∠NOB=∠DOC  ( vertically opposite angles)

And, ∠OBN = ∠ODC ( By the property of interior alternative angles by same transversal)

Thus, By the property of similarity,

$\triangle NOB\sim \triangle DOC$

Therefore,  $\frac{OB}{OD} = \frac{BN}{DC}$ = $\frac{BN}{2BN} = \frac{1}{2}$

Answer 2

In parallelogram ABCD,  the ratio BO:OD is 1:2

We have the ABCD is parallelogram, point M is the midpoint of the side BC , point O is the intersection point of the segment AM and diagonal BD.

What is the parallelogram?

A four-sided plane rectilinear figure with opposite sides parallel.

Let N is the mid point AB

Here DC is the diagonal of parallelogram and it is 2 times BN

Therefore we get,AN=NB or 2×BN=DC. .........( AB=DC)

We have join Points N and C.

In triangles NOB and DOC,∠NOB=∠DOC .....(1)(  opposite angles)

And, ∠OBN = ∠DOC .....................(2)( By the property of interior alternative angles )

Therefore from first and second we get the triangle NOB and triangle DOC are similar triangle.

Thus, By the property of similarity,

$\Delta NOB \sim \Delta DOC$$\frac{OB}{OD} =\frac{BN}{DC}=\frac{BN}{2BN}=\frac{1}{2}$

Therefore the ratio BO:OD is 1:2.

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