In ΔABC, if m ∠A = m∠C, m∠B = ß (where ß is an acute angle), and BC = x, which expression gives the length of b, the side opposite ∠B ?

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In ΔABC, if m ∠A = m∠C, m∠B = ß (where ß is an acute angle), and BC = x, which expression gives the length of b, the side opposite ∠B ?

Geography
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|dw:1444236085588:dw|

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|dw:1444289573188:dw| since we are told ∠A = ∠C we know that ABC is an issueless triangle where AB = BC = x so we get the above diagram. so now, to find side b, we can use the cosine rule which states \[c^2 = a^2 + b^2 - 2abcosC\]|dw:1444290019267:dw| so we can apply this to our question, to get, \[b^2 = x^2 + x^2 - 2(x)(x)\cosß\] which simplifies down to \[b = \sqrt{2x^2 - 2x^2\cosß}\] and you can factor the \[2x^2\] to the from to get \[b = \sqrt{2x^2(1 - \cosß)}\] ...Hope that made sense :)

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