Question

a certain triangle has a perimeter of 3064 mi. the shortest side measures 78 mi less than the middle side, and the longest side measures 373 mi more than the middle side. find the lengths of the three sides. the shortest side is □ mi long.

Explanation:

Step1: Define variables

Let the length of the middle side be \( x \) miles. Then the shortest side is \( x - 78 \) miles, and the longest side is \( x + 373 \) miles.

Step2: Set up the perimeter equation

The perimeter of a triangle is the sum of its three sides. So, \( (x - 78) + x + (x + 373) = 3064 \).

Step3: Simplify the equation

Combine like terms: \( 3x + 295 = 3064 \).

Step4: Solve for \( x \)

Subtract 295 from both sides: \( 3x = 3064 - 295 = 2769 \). Then divide by 3: \( x=\frac{2769}{3}=923 \).

Step5: Find the shortest side

The shortest side is \( x - 78 = 923 - 78 = 845 \).

Answer:

845