__Basic Proportionality Theorem Class 10th __
__(OR) B.P.T. or Thales Theorem__

__ ____Converse of basic proportionality theorem, thales theorem 10th standard, theorem 6.2 class 10__

**Statement:- **

**If a line is drawn parallel to one
side of the triangle to intersect the other two sides in two distinct points, the other two sides are divided in the same ratio.**

**Given:- **

**A Î” ABC in which line ***l ***॥**** BC, intersect side AB and AC at point D and E**

**To Prove :- **
**Construction :- **

**Draw EM ****ä¸„**** AB and DN ****ä¸„**** AC. Also join
BE and CD**

**Proof :- Area of triangle = ** ✕ Base ✕ Height

**Area of △ ADE = ** ✕ AD ✕ EM ..............(1)

**Area of △ BDE = ** ✕ BD ✕ EM ..............(2)

**Divide equation (1) by equation (2) we get**

**Similarly**

**Î”BED and Î”CDE are two triangles on the same base and lie
between the same parallel DE and BC**

** ****∴**** Ar(Î”BED) = Ar(Î”CDE)
.................. (5)**

**From equation (3), (4), (5) we get**

**Hence prove the Basic Proportionality Theorem.**

**Note:- ***For the examination point of view students should study the basic proportionality theorem only its converse is only a motivational theorem. *

__Converse of Basic Proportionality Theorem__

**If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. **

**Given:- ****In Triangle ABC, **
**To Prove :- ****Line ***DE*॥ BC

**Construction:-**

**If DE is not parallel to BC, then let us take another line DE'****॥**** BC**

**Proof:-**

**
**
**In ****Î”ABC, ****DE'****॥**** BC ****Therefore by B.P.T**

**Therefore **

**Adding 1 on both side**

**This is possible only if E and E' coincide with each other**

**⇒ E and E' represent the same point on the side of the triangle.**

** Hence DE is parallel to the side BC**

**Converse of BPT is proved**

**Important result based on BPT**

**If a line intersects side AB and AC of a ****Î”ABC at D and E respectively and is parallel to BC, then prove that **
**Solution :- **

**It is given that DE॥BC, therefore by BPT**

......... (∵ By Invertendo

**Adding 1 on both side we get**

.......... ( ∵ By Invertendo

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Thanks sir

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