Basic Proportionality Theorem (Thales Theorem)

A line is parallel to one side of triangle which intersects remaining two sides in two distinct points then that line divides sides in same proportion.

Given: In ΔABC line l || side BC and line l intersect side AB in P and side AC in Q.

To prove: `"AP"/"PB" = "AQ"/"QC"`

Construction: Draw CP and BQ

Proof: ΔAPQ and ΔPQB have equal height.

`("A"(Δ"APQ"))/("A"(Δ"PQB")) = (["______"])/"PB"`   .....(i)[areas in proportion of base]

`("A"(Δ"APQ"))/("A"(Δ"PQC")) = (["______"])/"QC"`  .......(ii)[areas in proportion of base]

ΔPQC and ΔPQB have [______] is common base.

Seg PQ || Seg BC, hence height of ΔAPQ and ΔPQB.

A(ΔPQC) = A(Δ______)    ......(iii)

`("A"(Δ"APQ"))/("A"(Δ"PQB")) = ("A"(Δ "______"))/("A"(Δ "______"))`   ......[(i), (ii), and (iii)]

`"AP"/"PB" = "AQ"/"QC"`   .......[(i) and (ii)]

D and E are points on the sides AB and AC respectively of a ΔABC such that DE║BC. Find the value of x, when

AD = 4cm, DB = (x – 4) cm, AE = 8cm and EC = (3x – 19) cm.  

In the given figure, PQ || AC. If BP = 4 cm, AP = 2.4 cm and BQ = 5 cm, then length of BC is ______.

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 4x − 3, AE = 8x – 7, BD = 3x – 1 and CE = 5x − 3, find the volume of x.

ΔABC ~ ΔDEF. If AB = 4 cm, BC = 3.5 cm, CA = 2.5 cm and DF = 7.5 cm, then the perimeter of ΔDEF is ______.

In figure, PA, QB, RC and SD are all perpendiculars to a line l, AB = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm. Find PQ, QR and RS.

ΔABC and ΔDBC lie on the same side of BC, as shown in the figure. From a point P on BC, PQ||AB and PR||BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR||AD. 

Construct an equilateral triangle of side 7 cm. Now, construct another triangle similar to the first triangle such that each of its sides are `5/7` times of the corresponding sides of the first triangle.