Question

Solve it !

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Maths Aryabhatta!

Both the questions.

Answer 1

2.(i) ABCD is a rectangle, it means
                         AB = CD
                         AC = BD
and AB||CD also, AC|| BD [because a rectangle is also a parallelogram]

again, ABEF is ||gm
so, AB = FE and AF = BE
also, AB||FE and AF||BE

because, AB||CD from rectangle,
               AB|| FE from ||gm,
so, CD||FE ------------------------(1)

again, AB = CD from rectangle,
           AB = FE from ||gm
so, CD = FE ----------------------------(2)

from equation (1) and (2) we can say that,
  CDEF is a parallelogram.
[note :-  any quadrilateral be a ||gm,
when two opposite sides will be equal and same opposite sides will be parallel ]

(ii) area of ||gm = base * altitude
now, area of quad. CDEF = base of CDEF *altitude of CDEF
                                         = CD *EQ [shown pic 1]
      area of ||gm ABEF = AB*EP
      area of rect. ABCD = length* breadth = CD*BD
because, CD = AB  and BD = PQ
then,  area of rect. ABCD = CD*PQ

now,RHS=  area of quad. CDEF = CD*EQ
              = CD*(EP + PQ)
              = CD*EP + CD*PQ
              = area of ||gm ABEF + area of rect. ABCD = LHS

3.(i) see 2nd attachment,
Let XO is || PQ and we know, diagonals of ||gm is bisected each other,
so, SO = OQ
hence, according to Thales theory,
XO is half of PQ,
e.g., XO = 1/2 PQ
but XO|| PQ so, we can say that XO||PM
so, XO = PM = 1/2PQ
hence, M is the midpoint of  PQ.
            L is the midpoint of SR.
hence, LM is bisected PQRS in two ||gm.
e.g., ar(PMLS) = ar(MQRL) = 1/2 ar(PQRS)-----------(1)

now, we know, a parallelogram is divided by four equal triangles.
e.g., ΔPOS = ΔSOR =ΔROQ = ΔQOP = 1/4 ||gm PQRS
from(1)
ΔPOS = ΔSOR =ΔROQ = ΔQOP = 1/2 ||gm PMLS ---------------(2)
hence, ΔPOS = 1/2 ||gm PMLS
  or, 2* ΔPOS = ||gm PMLS

(ii) from equation (2)
  ΔPOS = ΔQOR = 1/2 ||gm PMLS
  so, ΔPOS + ΔQOR = 1/2 ||gm PMLS  + 1/2||gm PMLS
                                 = ||gm PMLS
now, from equation (1)
ΔPOS + ΔQOR = 1/2 ||gm PQRS


(iii) from equations (2), we can easily, say that,
 ΔPOS  + ΔQOR = ΔPOQ + ΔSOR
                                    

Answer 2

$\underline{\underline{\Huge\mathfrak{Answer ;}}}$

Dear ,

• See the Attached file ⤴️


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