Question

in the figure triangle pqr is an equilateral triangle the point n is on the ray qr such that qr =rn prove pn sq. = 3 pr sq.

Answer 1

Answer:

Here triangle PRN is an isosceles triangle with PR = RN. So

angle RNP= angle PNR.

And from triangle PQR, anglePRN= 120 degrees{prq+prn=180}

In triangle PRN, PRN+PNR+RPN=180

                         120+2PNR=180

                         2PNR=60

                         PNR=30

IN TRIANGLE NPQ, ANGLE NPQ=90{NPQ=QPR+NPR=60+30=90}

SO TRIANGLE NPQ IS A RIGHT ANGLED TRIANGLE.

USING PYTHAGORAS THEOREM,

QNsqr=PQsqr+PNsqr

[2PR]sqr=PRsqr+PNsqr {PR=PQ AND QN=2PQ}

4PRsqr-PRsqr=PNsqr

therefore, PN sqr=3 PRsqr. PROVED

HOPE YOU UNDERSTAND

Answer 2

this is ur proof u can either apply this method or

2nd method:-

prove triangle pdr and pqr congruent by asa test

then u will get dr=qr bcs of csct and rest is same

but I guess 1st is more preferable