Question

(a) Sides RS, RT & median RU of a triangle RST are respectively proportional to sides LM, LN &
median LP of another triangle LMN. Prove that triangle RST similar to triangle LMN.​

Answer 1

Given :  Sides RS, RT & median RU of a triangle RST are respectively proportional to sides LM, LN &  median LP of another triangle LMN.

To Find :  Prove that triangle RST similar to triangle LMN.​

Solution :

Draw a line PQ || LM   &  UV || RS

using as P &  U are mid point Hence

a 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.

line joining the mid-point of two sides of a triangle is equal to half the length of the third side

PQ = LM/2

NQ = LQ = LN/2

and

UV = RS/2

TV = RV = RT/2

RS =  LM

=> RS/2 = LM/2

=> UV = PQ  = TV = NQ    

RT = LN

=> RT/2  = LN/2

=> RV = LQ

UV = PQ

RV = LQ

RU = LP

=>  ΔUVR  ≅ ΔPQL   ( SSS)  

Hence ∠UVR = ∠PQL

=> 180° - ∠UVR = 180° - ∠PQL

=> ∠UVT = ∠PQN      

in Δ UVT  & Δ PQN

UV = PQ

∠UVT = ∠PQN  

VT = QN

=> Δ UVT ≅ Δ PQN

=> UT = PN

=> 2 UT = 2 PN

=> ST = MN  ( as U  & P are mid point )

in Δ RST  & Δ LMN

RS = LM

RT = LN

ST = MN

Hence Δ RST  ≅ Δ LMN   ( SSS)

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