Asked by Jacob
                Quad ABCD with diagonal DB.  <adc=90, <c=91, <cbd=43. segment ad=segment dc=9. 
Which segment is longer ab or ad and why?
            
        Which segment is longer ab or ad and why?
Answers
                    Answered by
            Steve
            
    by the law of sines, the side opposite the larger angle has the larger length.
    
                    Answered by
            Jacob
            
    How are you solving for <abd.  Making assumptions?
    
                    Answered by
            Steve
            
    by law of sines,
bd/sin91 = 9/sin43
bd = 13.19
by law of cosines,
ab^2 = 9^2 + bd^2 - 2(9)(bd)cos44
= 81 + 174 - 170.84
ab = 9.17
Looks like ab > ad
    
bd/sin91 = 9/sin43
bd = 13.19
by law of cosines,
ab^2 = 9^2 + bd^2 - 2(9)(bd)cos44
= 81 + 174 - 170.84
ab = 9.17
Looks like ab > ad
                    Answered by
            Jacob
            
    Thank you steve, but something must be wrong with this problem.  We have not learned law of cosines or law of sines yet and the correct answer is ad?? Baffled at how to get this based on what we are studying, SAS and SSS inequality theorems.
    
                    Answered by
            Steve
            
    Hmmm. If <C were also 90, then DA would be parallel to BC. In that case triangles ADB and DCB would be congruent, and ABCD would be a square, with ad=ab. But that would also require <cbd to be 45, not 43.
So, since <cbd is less than 45, bc > ad, so also ab > ad.
I still don't get ad > ab.
Also, since SAS and SSS are for congruency, not sure how they apply in determining inequality here.
    
So, since <cbd is less than 45, bc > ad, so also ab > ad.
I still don't get ad > ab.
Also, since SAS and SSS are for congruency, not sure how they apply in determining inequality here.
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