## PROVING BASIC PROPORTIONALITY THEOREM IN GIVEN TRIANGLE

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.

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

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 