Understanding the Triangle and its Circumradius

The question asks for the length of the circumradius of a triangle where each side measures 12 cm. A triangle with all three sides of equal length is known as an equilateral triangle. Equilateral triangles have several special properties, including specific formulas for their area, height, inradius, and circumradius.

Identifying the Triangle Type

Given:

• Side length 1 = 12 cm
• Side length 2 = 12 cm
• Side length 3 = 12 cm

Since all sides are equal, the triangle is an equilateral triangle.

Circumradius of an Equilateral Triangle

The circumradius (\(\small R\)) of a triangle is the radius of the circle that passes through all three vertices of the triangle. This circle is called the circumcircle. For an equilateral triangle with side length \(\small a\), the formula for the circumradius is:

\(\small R = \frac{a}{\sqrt{3}}\)

Alternatively, you can also derive this from the general circumradius formula \(\small R = \frac{abc}{4K}\), where \(\small a, b, c\) are side lengths and \(\small K\) is the area. For an equilateral triangle with side \(\small a\), \(\small a=b=c\) and \(\small K = \frac{\sqrt{3}}{4} a^2\). So, \(\small R = \frac{a \cdot a \cdot a}{4 \cdot \frac{\sqrt{3}}{4} a^2} = \frac{a^3}{\sqrt{3} a^2} = \frac{a}{\sqrt{3}}\).

Calculating the Circumradius Length

Using the formula \(\small R = \frac{a}{\sqrt{3}}\) with the given side length \(\small a = 12\) cm:

\(\small R = \frac{12}{\sqrt{3}}\)

To rationalize the denominator, multiply the numerator and the denominator by \(\small \sqrt{3}\):

\(\small R = \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12\sqrt{3}}{3}\)

Now, simplify the expression:

\(\small R = 4\sqrt{3}\)

So, the length of the circumradius of the equilateral triangle is \(\small 4\sqrt{3}\) cm.

Comparing with Options

Let's compare our calculated circumradius with the given options:

• Option 1: \(\small 8\sqrt{3}\) cm
• Option 2: \(\small 2\sqrt{3}\) cm
• Option 3: \(\small 6\sqrt{3}\) cm
• Option 4: \(\small 4\sqrt{3}\) cm

Our calculated value \(\small 4\sqrt{3}\) cm matches Option 4.

Revision Table: Equilateral Triangle Properties

Additional Information on Triangle Centers

The circumcenter is the center of the circumcircle. It is the point where the perpendicular bisectors of the sides of the triangle intersect. In an equilateral triangle, the circumcenter coincides with several other important centers:

• Centroid: The intersection of the medians (lines from a vertex to the midpoint of the opposite side).
• Orthocenter: The intersection of the altitudes (perpendiculars from a vertex to the opposite side).
• Incenter: The intersection of the angle bisectors, and the center of the incircle (the circle tangent to all three sides).

In equilateral triangles, these four centers are all the same point. The ratio of the circumradius to the inradius (\(\small R:r\)) in an equilateral triangle is always \(\small 2:1\). In our case, \(\small R = 4\sqrt{3}\) cm, so the inradius would be \(\small r = \frac{R}{2} = \frac{4\sqrt{3}}{2} = 2\sqrt{3}\) cm. Using the inradius formula \(\small r = \frac{a}{2\sqrt{3}} = \frac{12}{2\sqrt{3}} = \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\) cm, this confirms the ratio.