# Basic Proportionality Theorem

Here we will learn how to prove the basic proportionality theorem with diagram.

A line drawn parallel to one side of a triangle divides the other two sides proportionally.

Given: In ∆XYZ, P and Q are points on XY and XZ respectively, such that PQ ∥ YZ.

To prove: $$\frac{XP}{PY}$$ = $$\frac{XQ}{QZ}$$.

Proof:

 Statement Reason 1. In ∆XYZ and ∆XPQ,(i) ∠YXZ = ∠PXQ(ii) ∠XYZ = ∠XPQ 1.(i) Common angle(ii) Corresponding angles 2. ∆XYZ ∼ ∆XPQ 2. AA criterion of similarity. 3. $$\frac{XY}{XP}$$ = $$\frac{XZ}{XQ}$$ 3. Corresponding sides of similar triangles are proportional. 4. $$\frac{XY}{XP}$$ – 1 = $$\frac{XZ}{XQ}$$ – 1⟹ $$\frac{XY - XP}{XP}$$ = $$\frac{XZ - XQ}{XQ}$$⟹ $$\frac{PY}{XP}$$ = $$\frac{QZ}{XQ}$$ 4. By subtracting 1 from both sides of statement 3. 5. $$\frac{XP}{PY}$$ = $$\frac{XQ}{QZ}$$ 5. Taking reciprocals of both sides in statement 4.

Solved examples using basic proportionality theorem:

1. If in a ∆XYZ, P and Q are two points on XY and XZ respectively such that XP = 4 cm, PY = 3 cm, XQ = = 6 cm, QZ = 4.5 cm and ∠XPQ = 40° then find ∠XYZ.

Solution:

Here, $$\frac{XP}{PY}$$ = $$\frac{4 cm}{3 cm}$$ = $$\frac{4}{3}$$, and

$$\frac{XQ}{QZ}$$ = $$\frac{6 cm}{4.5 cm}$$ = $$\frac{4}{3}$$

Therefore, $$\frac{XP}{PY}$$ = $$\frac{XQ}{QZ}$$

⟹ PQ ∥ YZ

Therefore,  ∠XYZ = ∠XPQ = 40°.

2. In the given figure, if XP = 6 cm, YP = 2 cm, XQ = 7.5 cm, find QZ.

Solution:

By basic proportionality theorem,

$$\frac{XP}{PY}$$ = $$\frac{XQ}{QZ}$$

⟹ $$\frac{6 cm}{2 cm}$$ = $$\frac{7.5 cm}{QZ}$$

⟹ QZ = $$\frac{7.5 cm × 2}{6}$$

⟹ QZ = 2.5 cm.

3. At a certain time of the day, a man, 6 feet tall, casts his shadow 8 feet long. Find the length of the shadow cast by a building 45 feet high, at the same time.

Solution:

Let the length of the shadow of the building be x.

As the source of light is the sun, XZ ∥ PQ and, hence ∆YXZ ∼ ∆YPQ.

Therefore, $$\frac{\textrm{Height of the Man}}{\textrm{Height of the Building}}$$ = $$\frac{\textrm{Length of Shadow Cast by the Man}}{\textrm{Length of Shadow Cast by the Building}}$$

⟹ $$\frac{6 ft}{45 ft}$$ = $$\frac{8 ft}{x}$$

⟹ x = 60 feet.