Basic Proportionality Theorem Proof

If a given line passes through the two sides of the given triangle and parallel to the third side, then it cuts the sides proportionally. This is called the Basic Proportionality theorem..

Basic Proportionality Theorem Proof

STATEMENT
A line parallel to one side of a triangle divides the other two sides into parts of equal proportion..

Given:

In triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively.

Image/Diagram:

To Prove:

AP/PB = AQ/QC

Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through each of these points on AB, draw lines parallel to BC to cut AC.

PROOF:

Step 1:
	Cut AP into equal parts and draw lines through these points parallel to BC
Step 2:
	By intercept theorem,AQ is cut into equal parts
Step 3:
	By Step 1,QC is cut into m equal parts
Step 4:
	By construction,AP/PB=l/m
Step 5:
	By step 2 and 3,AQ/QC=l/m
Step 6:
	By step 4 and 5,AP/PB=AQ/QC

Hence the proof..

If a given line passes through the two sides of the given triangle and parallel to the third side, then it cuts the sides proportionally. This is called the Basic Proportionality theorem..

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The basic proportionality theorem can be used for many purposes like measuring the height of the pole.