Final answer:
In triangle ABC with right angle at C and BL as the angle bisector of angle ABC, the ratio CL : AC can be found using the Angle Bisector Theorem. Substituting the given angle BAC of 30 degrees, we can solve for the ratio and find that CL : AC is 1 : 3.
Step-by-step explanation:
In triangle ABC, angle C is a right angle and BL is the angle bisector of angle ABC. We are given that angle BAC is 30 degrees. To find the ratio CL : AC, we can use the Angle Bisector Theorem. According to the theorem, the ratio of the lengths of the two segments created by an angle bisector is equal to the ratio of the lengths of the two sides opposite that angle. In this case, CL : AC is equal to BL : AB. Since BL is the angle bisector of angle ABC, we can conclude that BL divides side AB into two segments, and these two segments have the same ratio as the two sides opposite angle ABC. Therefore, the ratio CL : AC is the same as the ratio BL : AB.
Let's label the length of BL as x. Since BL is the angle bisector, we can divide side AB into two segments, where one segment is x and the other segment is AB - x. Using the properties of a right triangle, we can write the following equation:
x / (AB - x) = tan(30°)
Simplifying this equation, we get:
x = (AB - x) * tan(30°)
Expanding and rearranging the equation:
x = AB * tan(30°) - x * tan(30°)
x + x * tan(30°) = AB * tan(30°)
x(1 + tan(30°)) = AB * tan(30°)
x = (AB * tan(30°)) / (1 + tan(30°))
Now, we can substitute the value of x back into the ratio to find the value of CL : AC:
CL : AC = BL : AB
CL : AC = x : AB
CL : AC = ((AB * tan(30°)) / (1 + tan(30°))) : AB
CL : AC = tan(30°) / (1 + tan(30°))
Using the identity sin(30°) = 1/2 and cos(30°) = √3/2, we can simplify the ratio:
CL : AC = (1/2) / (1 + (1/2))
CL : AC = (1/2) / (3/2)
CL : AC = 1/3
Hence, the ratio CL : AC is 1 : 3.