Question

In the rectangle KLMN given here, KP and MQ are perpendicular lines on the diagonal LN. Prove that NP = LQ and KP = MQ.

Answer 1

In given figure of ractangle KLMN
KN=LM
KL=MN
angle K,L,M,N are each 90degree

properties of ractangle that opposite sides are equal .

in triangle KPN and LQM
angle KPN =angle LQM .....(given) KP and MQ are perpendicular ..........1
KN = LN. properties of ractangle.......2
and
half of angle L = half of angle N.....3
from 1 2 3 we get triangle KPN and LQM are coungrent

hance NP = LQ and KP = MQ

Answer 2

from figure,

angle  kpl=90

angle nqm=90

let

angle qnm be x

then

angle plk=angle qnm          

[kL and NM are parallel lines and NL is the  transversal then alternate interior angles are equal ]

we know that,

sum of all interior angles of a triangle is 180

i.e

angle pnm+angle qmn+angle nqm=180

qmn+x+90=180

qmn+ x=180-90

qmn=90-x

similarly

angle pkl +angle kpl +angle plk=180

pkl+90 +x=180

pkl=180-90-x

pkl=90-x

from figure

NM=KL          [opposite sides are equall in a rectangle]

from figure,

angle qnm= angle qlm

NM=LK

angle qmn= angle pkl

according to A.S.A congruency rule

      triangle qnm is congruent to triangle plk

from figure,

QN=QP+PN--->eq.1

AND

PL=QP+QL------>EQ.2

QN=PL   [C.P.C.T are equall]--------->EQ.3

from eq. 1,2 and 3

QP+PN=QP+QL

PN=QP-QP+QL

PN=QL

NP=LQ

FROM FIGURE;

KP=MQ               [C.P.C.T are equal]  .