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:

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.

9th Grade Math

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