**Find the Missing Side Length Using Basic Proportionality Theorem :**

In this section, we will learn how we find the missing side length using the concept 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.

**Question 1 :**

In the figure given below the sides DE and BC are parallel. Find the length of the side EC ?

**Solution :**

Let EC = x

Given that :

The sides DE and BC are parallel.

By using basic proportionality theorem, we have

(AD/DB) = (AE/EC) ----(1)

AD = 1.5 cm, DB = 3 cm, AE = 1 cm

By applying the known values in (1), we get

(1.5/3) = (1/x)

1.5 x = 3

x = 3/1.5

x = 2 cm

Hence, the length of side EC is 2 cm.

**Question 2 :**

In the figure given below the sides DE and BC are parallel. Find the length of the side AD ?

**Solution :**

Let AD = x

Given that :

The sides DE and BC are parallel.

By using basic proportionality theorem, we have

(AD/DB) = (AE/EC) ----(1)

AD = x DB = 7.2 cm AE = 1.8 cm EC = 5.4 cm

(x/7.2) = (1.8/5.4)

5.4 x = 1.8 x 7.2

x = (1.8 x 7.2)/5.4

x = 2.4 cm

**Question 3 :**

E and F are points on the sides PQ and PR respectively of a ∆ PQR. For each of the following cases, state whether EF ∥ QR.

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

(ii) PE = 4 cm, EQ = 4.5 cm, PF = 8 cm and FR = 9 cm

(iii) PQ = 1.28 cm PR = 2.56 cm PE = 0.18 cm and PF = 0.36 cm

**Solution :**

Given that :

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

(PE/EQ) = (PF/FR) ---(1)

(3.9/3) = (3.6/2.4)

1.3 ≠ 1.5

So, the sides EF and QR are not parallel.

(ii) PE = 4 cm , EQ = 4.5 cm, PF = 8 cm and FR = 9 cm

**Solution :**

According to basic proportionality theorem, we have

(PE/EQ) = (PF/FR)

(4/4.5) = (8/9)

0.88 = 0.88

So, the sides EF and QR are parallel.

(iii) PQ = 1.28 cm PR = 2.56 cm PE = 0.18 cm and PF = 0.36 cm

**Solution :**

PQ = PE + EQ

1.28 = 0.18 + EQ

EQ = 1.28 - 0.18 = 1.1

PR = PF + FR

2.56 = 0.36 + FR

2.56 - 0.36 = FR

FR = 2.2

According to basic proportionality theorem, we have

(PE/EQ) = (PF/FR)

(0.18/1.1) = (0.36/2.2)

0.1636 = 0.1636

So, EF is parallel to QR.

After having gone through the stuff given above, we hope that the students would have understood, how to find the missing side length using basic proportionality theorem.

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