Question

Consider ABC with the measure of angle B equal to 60 degrees, and side lengths a=4 and c=5. Which option lists an expression that is equivalent to the length of side b?

Answer 1

`b=√(25+16-20)`

Explanation:

(The choices are attached)

We don't if the triangle ABC is a right triangle of other kind, that's why we need to use the cosines law which is defined as

`b^(2)=a^(2) +c^(2)-2.a.c.cosB\°`

Where
`a=4`
`c=5` and
`B=60\°`, replacing these values we have

`b^(2)=4^(2)+5^(2)-2(4)(5)cos(60\°)\\ b=\sqrt{16+25-40(1)/(2) }\\ b=√(25+16-20)`

Remember that
`cos60\°=(1)/(2)`

Therefore, the second choice is the correct answer

`b=√(25+16-20)`

Answer 2

For this case we can use the law of cosine to determine the length of side b.
We have then:
b ^ 2 = a ^ 2 + c ^ 2 - 2 * a * c * cos (B)
Substituting values we have:
b ^ 2 = 4 ^ 2 + 5 ^ 2 - 2 * 4 * 5 * cos (60)
Clearing b:
b = root (4 ^ 2 + 5 ^ 2 - 2 * 4 * 5 * cos (60))
Answer:
an expression that is equivalent to the length of side b is:
b = root (4 ^ 2 + 5 ^ 2 - 2 * 4 * 5 * cos (60))