Final answer:
To find the ratio of the longest side to the perimeter of a triangle with angles in the ratio 4:1:1, determine the measures of the angles, find the lengths of the sides using the law of sines, and the ratio is 4:6. So, the correct answer is option B.
Step-by-step explanation:
To find the ratio of the longest side to the perimeter of a triangle whose angles are in the ratio 4:1:1, we first need to find the measures of the angles.
The sum of the angles in a triangle is 180 degrees, so the three angles would be 4x, x, and x, where x is a common factor.
Since 4x + x + x = 180, we can solve for x: 6x = 180, x = 30.
The three angles would be 120°, 30°, and 30°.
Next, we need to determine the lengths of the sides.
Let's assume the longest side has a length of 4y, and the other two sides have lengths of y each.
Using the law of sines, we can set up the following proportion: sin(120°)/4y = sin(30°)/y.
Solving for y, we get y = 2√3. Therefore, the longest side is 4y = 8√3, and
the perimeter is 4y + y + y = 6y = 6(2√3)
= 12√3.
Finally, we can find the ratio of the longest side to the perimeter: (8√3)/(12√3) = 2/3.
So, the correct answer is option B. 4:6.