In a quadrilateral ABCD ,which is not a trapezium.It is known that <DAB=<ABC=60 DEGREE.moreover , <CAB=<CBD.Then A) AB=BC+CD B) AB=AD+CD C) AB=BC+AD D) AB=AC+AD ? Choose correct option.
2 Answers
C)
Explanation:
To ease the comprehension of the problem refer to the figure below:
If the sides BC and AD are extended the
First, in addition to the angles informed it was possible to determine other interior angles of the triangles ABC and ABD, as shown in the figure.
Then using the Law of Sines on these triangles:
#triangle_(ABC)#
#"AB"/sin (120^@alpha)="BC"/sin alpha="AC"/sin 60^@#
#triangle_(ABD)#
#"AB"/sin (60^@+alpha)="AD"/sin (60^@alpha)="BD"/sin 60^@#
To focus on viable hypotheses I suggest to begin trying to prove a specific case.
For instance
From
From
Note that
Finally,
#AD+BC=5*(sin 15^@+sin 45^@)/(sin 75^@)=5*1=AB#
So in this case hypothesis (C) is true (I tried the others and verified that they are false.)
Now for the general proof of the hypothesis (C), following the same steps that worked for the specific proof:
From
Observation:
#sin (120^@alpha)=sin (180^@(120^@alpha))=sin(60^@+alpha)#
Therefore,
# "BC" ="AB" * (sin alpha)/((sqrt(3)/2)*cos alpha+(1/2)*sin alpha)#
From
This will get you
#AD="AB"*sin(60^@alpha)/sin (60^@+alpha)#
#=AB((sqrt(3)/2)*cos alpha(1/2)*sin alpha)/((sqrt(3)/2)*cos alpha+(1/2)*sin alpha)#
Finally,
#BC+AD="AB"(sin alpha+(sqrt(3)/2)*cos alpha(1/2)*sin alpha)/((sqrt(3)/2)*cos alpha+(1/2)*sin alpha)#
#BC + AD ="AB" * 1#
Therefore,
#AB=BC+AD# and the hypothesis (C) is true.**
Option( c)is correct
i.e.AB=BC+AD
Explanation:
Given

#"In quadrilateral "ABCD, /_DAB=/_ABC=60^@# 
#/_CAB=/_CBD#
Construction

# BC and AD " are produced to intersect at E"# 
# In Delta ABE,/_EBA=/_EAB=60^@ =>/_BEA=60^@# 
#:. Delta ABE " is equilateral">AB=BE=EA#
Now
#In DeltaABC and Delta BED, /_BED=/_CBA=60^@#
#/_EBD=/_CAB (given)and BE= AB (proved)# # DeltaABC and Delta BED " are congruent by ASA Rule"# #:.DE = BC " corresponding sides"#
#"of congruent triangles"# # "Finally", AB=AE=AD+DE=AD+BC#
So option (c) is correct