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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.

Basic Proportionality Theorem

To prove: XPPY = XQQZ.

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. XYXP = XZXQ

3. Corresponding sides of similar triangles are proportional.

4. XYXP – 1 = XZXQ – 1

XYXPXP = XZXQXQ

PYXP = QZXQ

4. By subtracting 1 from both sides of statement 3.

5. XPPY = XQQZ

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.

Problems on Basic Proportionality Theorem

Solution:

Here, XPPY = 4cm3cm = 43, and

XQQZ = 6cm4.5cm = 43

Therefore, XPPY = XQQZ

⟹ PQ ∥ YZ

Therefore,  ∠XYZ = ∠XPQ = 40°.


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

Numerical Problems on Basic Proportionality Theorem

Solution:

By basic proportionality theorem,

XPPY = XQQZ

6cm2cm = 7.5cmQZ

⟹ QZ = 7.5cm×26

⟹ 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.

Basic Proportionality Theorem Problem

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

Therefore, Height of the ManHeight of the Building = Length of Shadow Cast by the ManLength of Shadow Cast by the Building

⟹ 6ft45ft = 8ftx

⟹ x = 60 feet.







9th Grade Math

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