A triangle has a perimeter of 161 miles. The length of each of the two shorter sides of the triangle is two thirds the length of the longest side. Find the length of each side of the triangle.

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Let's denote the length of the longest side of the triangle as "x" miles. According to the given information, the length of each of the two shorter sides is two-thirds of the longest side.

So, the length of the first shorter side is (2/3) * x = (2x)/3 miles.
The length of the second shorter side is also (2/3) * x = (2x)/3 miles.

The perimeter of a triangle is the sum of the lengths of all three sides. In this case, the perimeter is given as 161 miles. Therefore, we can write the equation:

x + (2x)/3 + (2x)/3 = 161

To solve for x, we can simplify the equation:

(3x + 2x + 2x)/3 = 161
(7x)/3 = 161
7x = 3 * 161
7x = 483
x = 483 / 7
x ≈ 69

So, the longest side of the triangle is approximately 69 miles.

Now, let's substitute this value back into the expressions for the shorter sides to find their lengths:

First shorter side = (2x)/3 ≈ (2 * 69)/3 ≈ 138/3 ≈ 46 miles

Second shorter side = (2x)/3 ≈ (2 * 69)/3 ≈ 138/3 ≈ 46 miles

Therefore, the length of each side of the triangle is approximately:
Longest side = 69 miles
Shorter sides = 46 miles each

To solve this problem, let's denote the length of the longest side as x. According to the information given, the length of the two shorter sides is two-thirds of the longest side. So, the length of each of the two shorter sides can be expressed as (2/3)x.

The perimeter of a triangle is the sum of the lengths of all three sides. In this case, the perimeter is given as 161 miles. So, we can form the following equation:

x + (2/3)x + (2/3)x = 161

Now, let's simplify the equation:

x + (4/3)x + (4/3)x = 161

Combining like terms:

x + (8/3)x = 161

Multiplying both sides of the equation by 3 to get rid of the fraction:

3x + 8x = 483

11x = 483

Dividing both sides by 11:

x = 483 / 11

Calculate the value of x:

x ≈ 43.909

Now, we can find the length of each side of the triangle:

Longest side: x ≈ 43.909 miles
Shorter sides: (2/3)x ≈ (2/3)(43.909) ≈ 29.273 miles

Therefore, the lengths of the sides of the triangle are approximately 43.909 miles, 29.273 miles, and 29.273 miles.

let x = longest side, then 2/3x = shorter sides.

x + 2(2/3x) = 161

Solve for x, then 2/3 x.