In a right triangle, we can use trigonometric functions to find the lengths of sides based on angles. Here, we have a right triangle ABC where angle B measures 60 degrees, and side BC is 18.
To find the length of AC, we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Since angle B is 60 degrees and side BC is opposite this angle, we can set up the following relationship:
sin(B) = Opposite / Hypotenuse
In our triangle, that translates to:
sin(60°) = BC / AC
Here, BC (the side opposite angle B) is 18, and AC is the hypotenuse. We know that:
sin(60°) = √3/2
Now we can substitute the known values into the equation:
√3/2 = 18 / AC
To solve for AC, we can rearrange the equation:
AC = 18 / (√3/2)
Multiplying both sides by 2 to eliminate the fraction gives:
AC = 36 / √3
To simplify this expression, we can multiply the numerator and the denominator by √3:
AC = (36√3) / 3
Therefore:
AC = 12√3
This value, approximately equal to 20.78 units, is the length of AC. So, in conclusion, given the triangle ABC with the specifications provided, the length of AC is 12√3.