Question

1. In ABC, A=45 degrees, B=75 degrees, and BC=12. What is the length of AB?

Answer 1

We know that sum of the interior angles of any triangle equal 180° :

So :

A + B + C = 180°

45° + 75° + C = 180°

120° + C = 180°

Both sides minus ( 120°) :

C = 180° - 120°

C = 60°

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According to the theorem of sinuses :

$\frac{AB}{ \sin(C) } = \frac{BC}{ \sin(A) }$

$\frac{AB}{ \sin(60) } = \frac{12}{ \sin(45) }$

$\frac{AB}{ \frac{ \sqrt{3} }{2} } = \frac{12}{ \frac{ \sqrt{2} }{2} }$

$Multiply \: both \: sides \: by \: \frac{ \sqrt{3} }{2}$

$AB = \frac{24}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2}$

$AB = \frac{12 \sqrt{3} }{ \sqrt{2} }$

$AB = \frac{2 \times 6 \sqrt{3} }{ \sqrt{2} }$

$AB = \frac{ ({ \sqrt{2} })^{2} \times 6 \sqrt{3} }{ \sqrt{2} }$

$AB = \frac{ \sqrt{2} \times \sqrt{2} \times 6 \sqrt{3} }{ \sqrt{2} }$

$AB = \frac{ \sqrt{2} \times 6 \sqrt{3} }{1}$

$AB = \sqrt{2} \times 6 \sqrt{3}$

$AB = 6 \sqrt{2 \times 3}$

$AB = 6 \sqrt{6}$

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I think this is the correct answer.

And we're done.

Thanks for watching buddy good luck.

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