Can you see any SIgNs

Geometry Level 2

The sides of a triangle are a , b , c a,b,c and the angles opposite to them are A , B , C A, B, C respectively. Given:

a = 7 , b = 8 , sin ⁡ ( A ) = 8 9 a = 7, b=8, \sin{(A)} = \dfrac{8}{9}

what is the number of triangles that can be formed from the given data?


The answer is 0.

Mahdi Raza by Mahdi Raza , 16, India

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1 solution

Mahdi Raza
May 30, 2020

sin ⁡ ( A ) a = sin ⁡ ( B ) b sin ⁡ ( A ) ⋅ b a = sin ⁡ ( B ) sin ⁡ ( B ) = 8 9 ⋅ 8 7 sin ⁡ ( B ) = 64 63 > 1 \begin{aligned} \dfrac{\sin{(A)}}{a} &= \dfrac{\sin{(B)}}{b} \\ \\ {\sin{(A)}} \cdot \dfrac{b}{a} &= \sin{(B)} \\ \\ \sin{(B)} &= \dfrac{8}{9} \cdot \dfrac{8}{7} \\ \\ \sin{(B)} &= \dfrac{64}{63} \color{#3D99F6}{>1} \end{aligned}

Since sin ⁡ ( B ) ≯ 1 there exists no such triangle \boxed{\text{Since } {\color{#3D99F6}{\sin{(B)} \not> 1}} \text{ there exists no such triangle}}

3 Helpful 1 Interesting 1 Brilliant 0 Confused

If you are given the values of a, b, and sin A, by the law of sines the number of triangles must be either 0, 1, or 2. Given Brilliant's format of three tries before getting marked incorrect, you can cheat the system and just try 0, then 1, then 2 without even calculating the answer beforehand and be assured that one of those will be correct! (Or you could change the format to multiple choice so people can't do that.)

David Vreken - 1 year ago

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Ok, yes I agree. Thank you for enlightening me. Do you know how I can change it to multiple-choice? I think answers can't be changed by me, but moderators can. And I only know Mr. Chew as a moderator, but can't tag him, can you?. Thanks again!!

Edit: @Brilliant Mathematics

Mahdi Raza - 1 year ago

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For some reason I can't tag him, either.

David Vreken - 1 year ago

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@David Vreken – Hmm... Should I re-post the problem? Or should I try to tag someone else?

Mahdi Raza - 1 year ago

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@Mahdi Raza – If it were me I would just leave it as is. It would be nice if it could be changed to multiple choice, but probably not many people know that that 0, 1, and 2 can be the only answers, anyway.

David Vreken - 1 year ago

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@David Vreken – Yeah, I was thinking about that a bit. Because if someone knows the answer is either 0, 1 or 2, they might know the technique as well on how to solve. I think it will be preffered if the staff can change it, but it won't be a big deal if it isn't. Thank you once again, have a nice day!

Mahdi Raza - 1 year ago

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