Question

1. the ratio of the measures of the sides of a triangle is 9:7:3. if the perimeter of the triangle is 266 inches, find the length of the shortest side. show all work.

Explanation:

Step1: Let the sides be \(9x\), \(7x\), \(3x\)

Since the ratio of the sides is \(9:7:3\), we can represent the sides as \(9x\), \(7x\), and \(3x\) where \(x\) is a common multiplier.

Step2: Use the perimeter formula

The perimeter \(P\) of a triangle is \(P=a + b + c\). Here, \(P=266\) inches and \(a = 9x\), \(b = 7x\), \(c = 3x\). So, \(9x+7x + 3x=266\).
Combining like - terms, we get \((9 + 7+3)x=266\), which simplifies to \(19x=266\).

Step3: Solve for \(x\)

Dividing both sides of the equation \(19x = 266\) by \(19\), we have \(x=\frac{266}{19}=14\).

Step4: Find the length of the shortest side

The shortest side is \(3x\). Substituting \(x = 14\) into \(3x\), we get \(3\times14 = 42\) inches.

Answer:

The length of the shortest side is \(42\) inches.