Examples of Basic Proportionality Theorem :

In this section, let us learn the basic proportionality theorem.

**Basic Proportionality Theorem :**

If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

**Converse of Basic Proportionality Theorem Examples :**

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 AP = 3 cm, AR = 4.5 cm, AQ = 6 cm, AB = 5 cm and AC = 10 cm. Find the length of AD.

**Solution :**

From the given information we get, (AB/AP) = (AC/AQ)

In triangle ABC,

(AB/AP) = (AC/AQ)

By using converse of “Thales theorem” PQ is parallel to BC.

RD = x

In triangle ABD, PR is parallel to BD

AD = AR + RD

AD = 4.5 + x

(AB/AP) = (AD/AR)

(5/3) = (4.5 + x)/4.5

(5 ⋅ 4.5)/3 = 4.5 + x

7.5 = 4.5 + x

x = 7.5 – 4.5

x = 3

Here we need to find the length of AD = 4.5 + x

= 4.5 + 3

= 7.5 cm

**Example 2 :**

E and F are points on the sides PQ and PR respectively, of a triangle PQR. For each of the following cases. Verify EF is parallel to QR.

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

**Solution :**

First let us draw the picture for the above details

To verify whether EF is parallel to QR we have to check the condition

(PE/EQ) = (PF/FR)

(3.9/3) = (3.6/2.4)

1.3 ≠ 1.5

So the sides EF and QR are not parallel.

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

**Solution :**

First let us draw the picture for the above details

To verify whether EF is parallel to QR we have to check the condition

(PE/EQ) = (PF/FR)

(4/4.5) = (8/9)

0.88 = 0.88

So the sides EF and QR are parallel.

**Example 3 :**

In the figure, AC is parallel to BD and CF is parallel to DF, if OA = 12 cm, AB = 9 cm, OC = 8 cm and EF= 4.5 cm, then find FQ.

**Solution :**

In triangle OBD, AC is parallel to BD. By using “Thales theorem” we get,

(OA/AB) = (OC/CD)

(12/9) = (8/CD)

CD = (8 **x**** **9)/12

= 72/12

= 6 cm

In triangle ODF, the sides CE and DF are parallel, by using “Thales theorem” we get,

(OC/CD) = (OE/EF)

(8/6) = (OE/4.5)

OE = (8 ⋅ 4.5)/6

= 36/6

= 6 cm

So, OF = OE + EF

= 6 + 4.5

= 10.5 cm

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

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