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If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the perimeter is

A. 2 : 7
B. 4 : 6
C. 1 : 6
D. 4 : 1

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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.

User Ulli Schmid
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