Given a triangle abc such that <abc=60° and <BAC=30°.calculate the value of /BC/,if /AC/=6cm
This is a 30, 60, 90 right triangle with the right angle at C
tangent 30 = BC/AC = BC/6 = 1/sqrt 3
To find the length of side BC in triangle ABC, we can use the law of sines. The law of sines states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant.
In triangle ABC, we have the following information:
- Angle BAC = 30 degrees
- Angle ABC = 60 degrees (opposite side BC)
- Side AC = 6 cm (opposite angle BAC)
We need to find side BC.
Using the law of sines, we can set up the following ratio:
sin(BAC) / BC = sin(ABC) / AC
Substituting the given values:
sin(30°) / BC = sin(60°) / 6cm
We know that sin(30°) = 1/2 and sin(60°) = √3/2.
Simplifying the equation further:
(1/2) / BC = (√3/2) / 6cm
To isolate BC, we can cross-multiply and solve for BC:
BC = (1/2) * (6cm) / (√3/2)
Simplifying further by dividing the numerator by the denominator:
BC = (6cm) / (√3)
To rationalize the denominator, we multiply the numerator and denominator by √3:
BC = (6cm * √3) / (√3 * √3)
Simplifying the denominator:
BC = (6cm * √3) / 3
Finally, simplifying the expression:
BC = 2cm * √3
Therefore, the value of BC is 2cm multiplied by the square root of 3, which is approximately 3.46 cm.