Question

the perimeter of an equilateral triangle is 876 centimeters. the three vertices of the triangle lie on a circle. the radius of the circle is w√3 centimeters. what is the value of w?

Explanation:

Step1: Find side - length of equilateral triangle

The perimeter of an equilateral triangle is \(P = 3s\), where \(s\) is the side - length. Given \(P=876\) cm, then \(s=\frac{876}{3}=292\) cm.

Step2: Use the relationship between an equilateral triangle and circum - circle

For an equilateral triangle with side - length \(s\) and circum - radius \(R\), the formula is \(s = \sqrt{3}R\). Here, the radius of the circle \(R = w\sqrt{3}\), and \(s = 292\) cm. Substituting \(R = w\sqrt{3}\) into \(s=\sqrt{3}R\), we get \(s=\sqrt{3}\times w\sqrt{3}=3w\).

Step3: Solve for \(w\)

Since \(s = 3w\) and \(s = 292\) cm, then \(w=\frac{292}{3}\) cm.

Answer:

\(\frac{292}{3}\)