Q. 75.0( 1 Vote )

In ΔPQR, PQ > PR; Lets cut off the line segment PS equal to the length of PR from the side PQ. Let’s join two points R and S. Let’s prove that

(i) PSR = 1/2 ( PQR + PRQ)

(ii) QRS = 1/2 ( PRQ - PQR)

Answer :

Given: ΔPQR, PQ > PR, PS = PR


To prove: (i)


(ii)


The figure to the given question is as shown below,



(i) Now Consider the ΔPSR,


We know in a triangle the sum of all three interior angles is equal to 180°.


So in this case,


PSR + PRS + P = 180°……(i)


But given PS = PR


And we know angles opposite to equal sides are equal, so


PSR = PRS………(ii)


Substituting this value in equation (i), we get


PSR + PSR + P = 180°


2PSR + P = 180°


P = 180° - 2PSR……(iii)


Now Consider the ΔPQR,


We know in a triangle the sum of all three interior angles is equal to 180°.


So in this case,


PQR + PRQ + P = 180°


Substituting the value of P from equation (iii), we get


PQR + PRQ + 180° - 2PSR = 180°


2PSR = PQR + PRQ + 180° - 180°


2PSR = PQR + PRQ


(iv)


Hence proved


(ii) From equation (ii), PSR = PRS


And from equation (iv),


Comparing these two we get



From figure, PRS = PRQ - QRS


Substituting this value in equation (v), we get







Hence Proved


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