**Converse of Basic Proportionality Theorem Examples :**

In this section, we see example problems based on converse of basic proportionality theorem.

**Converse of basic proportionality theorem :**

If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Given :

In triangle ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC.

**Example 1 :**

In the figure given below, A, B and C are points on OP, OQ and OR respectively such that AB ∥ PQ and AC ∥ PR. Show that BC ∥ QR.

**Solution :**

In triangle OPQ,

Given that :

AB is parallel to PQ

(OA/AP) = (OB/BQ) ---(1)

In triangle OPR,

Given that :

AC is parallel to PR

(OA/AP) = (OC/CR) ---(2)

(1) = (2)

(OB/BQ) = (OC/CR)

Hence the sides BC and QR are parallel.

**Example 2 :**

ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO/BO = OC/OD.

**Solution :**

Draw OM parallel to AB meeting BC at M.

In triangle ACB,

Given that :

OM is parallel to AB

(OC/OA) = (CM/MB) ----(1)

In triangle BDC,

OM is parallel to CD

(BM/MC) = (OB/OD)

By taking reciprocals on both sides, we get

(CM/MB) = (OD/OB) ----(2)

(OC/OA) = (OD/OB)

(OC/OD) = (OA/OB)

Hence proved.

**Example 3 :**

The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/OD. Show that ABCD is a trapezium

**Solution :**

AD = DB

AD/BD = 1 ---- (i)

Also, E is the mid-point of AC (Given)

AE = EC

AE/EC = 1 [From equation (i)]

From equation (i) and (ii), we get

AD/BD = AE/EC

By using the converse of basic proportionality theorem, the sides DE and BC are parallel.

**Example 4 :**

ABCD is a trapezium in which AB ∥ DC and its diagonals intersect each other at the point O. Show that (AO/BO) = (CO/DO)

In ΔADC, we have OE || DC

AE/ED = AO/CO ...**(i)**

In ΔABD, we have OE || AB

DE/EA = DO/BO ...**(ii)**

From equation **(i)** and **(ii)**, we get

AO/CO = BO/DO

AO/BO = CO/DO

After having gone through the stuff given above, we hope that the students would have understood, converse of basic proportionality theorem.

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