**Basic Proportionality Theorem :**

If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

**Converse of Basic Proportionality Theorem Examples :**

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

Given :

In triangle ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC.

**Example 1 :**

ABCD is a
quadrilateral with AB parallel to CD. A line drawn parallel to AB meets AD at P
and BC at Q. Prove that (AP/PD) = (BQ/QC)

**Solution :**

Join BD by intersecting the line PQ at the point Q.

In triangle DAB, PE and AB are parallel, by using “Thales theorem”

(AP/PD) = (BE/ED) ------- (1)

In triangle BCD EQ and DC are parallel, by using “Thales theorem”

(BE/ED) = ( BQ/QC) ------- (2)

(1) = (2)

(AP/PD) = ( BQ/QC)

**Example 2 :**

In t he figure, PC and QK are parallel BC and HK are parallel, if AQ = 6 cm, QH = 4 cm, HP = 5 cm, KC = 18 cm, then find AK and PB.

**Solution :**

In triangle APC, the sides PC and QK are parallel

By using “Thales theorem” we get

(AQ/QP) = (AK/KC)

QP = QH + HP

= 4 + 5 = 9 cm

(6/9) = (AK/18)

AK = (6 ⋅ 18)/9

AK = 12 cm

In triangle ABC, the sides BC and HK are parallel,

By using “Thales theorem” we get

(AH/HB) = (AK/KC)

AH = AQ + QH

= 6 + 4

= 10

(10/HB) = (12/18)

(10 **x** 18)/12 = HB

HB = 15 cm

Now we need to find the length of PB,

PB = HB – HP

= 15 – 5

= 10 cm

**Example 3 :**

In the figure DE is parallel to AQ and DF is parallel to AR prove that EF is parallel to QR.

**Solution :**

In triangle PQA, the sides DE is parallel to the side AQ

By using “Thales theorem” we get

(PE/EQ) = (PD/DA) ------ (1)

In triangle PAR, the sides DF is parallel to the side AR

By using “Thales theorem” we get

(PD/DA) = (PF/FR) ------ (2)

(1) = (2)

(PE/EQ) = (PF/FR)

From this we can decide EF is parallel to QR in the given triangle PQR

**Example 4 :**

In the figure the sides DE and AB are parallel and DF and AC are parallel. Prove that EF and BC are parallel.

**Solution :**

In triangle APB, the sides DE and AB are parallel

(PD/DA) = (PE/EB) ----- (1)

In triangle PAC, the sides DF and AC are parallel

(PD/DA) = (PF/FC) ----- (2)

(1) = (2)

(PE/EB) = (PF/FC)

Hence the sides EF and BC are parallel.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**