Question

is there any genius to solve this question???????
solve it and get 44 pts
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Answer 1

GIVEN : ABCD is a ||gm , in which P is a point on AB such that AP : PB = 3 : 2 and Q on CD such that CQ : QD = 4 : 1.

TO PROVE : AR = 3/7AC

PROOF : Since ABCD is a ||gm , then AB || CD and AD || BC.

Since AB || CD and AC is a transversal, so

∠PAR = ∠QCR (Alternate interior angles) .........(1)

Since ABCD is a ||gm , then AB = CD and AD = BC , as opposite sides of ||gm are equal.

Let AB = CD = x

so, AP = 3x/5 and PB = 2x/5 [as, AP:PB = 3:2]and CQ = 4x/5 and QD = x/5 [as, CQ:QD = 4:1]

In △CQR and △APR, ∠QCR = ∠PAR [using (1)]∠QRC = ∠PRA [vertically opposite angles]⇒△CQR ~ △APR [AA similarity]

⇒CR/AR = CQ/AP=QR/PR (corresponding sides of similar △'s are proportional)⇒CR/AR = CQ/AP⇒CR/AR =4x/5=3x/5⇒CR/AR =4/3⇒CR/AR + 1= 4/3 + 1⇒CR + AR/AR = 7/3⇒AC/AR = 7/3⇒7AR = 3AC⇒AR = 3/7AC

Answer 2

Answer:GIVEN : ABCD is a ||gm , in which P is a point on AB such that AP : PB = 3 : 2 and Q on CD such that CQ : QD = 4 : 1.

TO PROVE : AR = 3/7AC

PROOF : Since ABCD is a ||gm , then AB || CD and AD || BC.

Since AB || CD and AC is a transversal, so

∠PAR = ∠QCR (Alternate interior angles) .........(1)

Since ABCD is a ||gm , then AB = CD and AD = BC , as opposite sides of ||gm are equal.

Let AB = CD = x

so, AP = 3x/5 and PB = 2x/5 [as, AP:PB = 3:2]and CQ = 4x/5 and QD = x/5 [as, CQ:QD = 4:1]

In △CQR and △APR, ∠QCR = ∠PAR [using (1)]∠QRC = ∠PRA [vertically opposite angles]⇒△CQR ~ △APR [AA similarity]

⇒CR/AR = CQ/AP=QR/PR (corresponding sides of similar △'s are proportional)⇒CR/AR = CQ/AP⇒CR/AR =4x/5=3x/5⇒CR/AR =4/3⇒CR/AR + 1= 4/3 + 1⇒CR + AR/AR = 7/3⇒AC/AR = 7/3⇒7AR = 3AC⇒AR = 3/7AC