The height of the equilateral triangle is 9 cm. What is the radius (in cm.) of the circle circumscribing the three vertices?

Options

A. 3
B. 6
C. 9
D. 12

Answer

To find the radius of the circumcircle of an equilateral triangle given its height, we can follow these steps:

### Step 1: Understand the relationship between the height and the side length of the equilateral triangle.
For an equilateral triangle, the height (h) can be expressed in terms of the side length (a) using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]

### Step 2: Rearrange the formula to find the side length.
Given that the height \( h = 9 \) cm, we can rearrange the formula to solve for \( a \):
\[ a = \frac{2h}{\sqrt{3}} \]
Substituting the value of \( h \):
\[ a = \frac{2 \times 9}{\sqrt{3}} = \frac{18}{\sqrt{3}} \]

### Step 3: Simplify the side length.
To simplify \( \frac{18}{\sqrt{3}} \), we can multiply the numerator and the denominator by \( \sqrt{3} \):
\[ a = \frac{18\sqrt{3}}{3} = 6\sqrt{3} \]

### Step 4: Use the formula for the circumradius of an equilateral triangle.
The circumradius (R) of an equilateral triangle can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]
Substituting the value of \( a \):
\[ R = \frac{6\sqrt{3}}{\sqrt{3}} = 6 \]

### Final Answer:
The radius of the circle circumscribing the three vertices of the equilateral triangle is \( 6 \) cm.

---