Understanding the Circumcenter and Altitude of an Equilateral Triangle

In an equilateral triangle, several important points coincide. The circumcenter, which is the center of the circle passing through all three vertices, is the same point as the incenter (center of the inscribed circle), the centroid (intersection of medians), and the orthocenter (intersection of altitudes). This central point lies on each median, altitude, and angle bisector of the triangle.

The Centroid Divides the Altitude

A key property of the centroid (which is also the circumcenter in an equilateral triangle) is that it divides each median (and thus each altitude) in a specific ratio. The centroid divides the median in the ratio 2:1, measured from the vertex. This means the distance from the vertex to the centroid is twice the distance from the centroid to the base (or the midpoint of the opposite side).

Relating Circumcenter Distance to Altitude Length

The problem states that the circumcenter is at a distance of 3.2 cm from the base of the equilateral triangle. Since the circumcenter is also the centroid, this distance represents the shorter segment of the altitude, which corresponds to the '1' part of the 2:1 ratio. Let the distance from the circumcenter to the base be $d$. We are given $d = 3.2$ cm.

Let the distance from the vertex to the circumcenter be $D$. According to the 2:1 ratio property, $D = 2 \times d$.

The total length of the altitude, let's call it $h$, is the sum of these two segments:

$\text{Altitude } h = D + d$

Substituting $D = 2d$, we get:

$h = 2d + d = 3d$

Calculating the Altitude Length

We are given $d = 3.2$ cm. Using the formula $h = 3d$, we can calculate the length of the altitude:

$h = 3 \times 3.2 \text{ cm}$

$h = 9.6 \text{ cm}$

Therefore, the length of each altitude of the equilateral triangle is 9.6 cm.

Verification using Properties

If the distance from the circumcenter to the base is 3.2 cm, this is the segment from the centroid to the base, which is 1/3 of the total altitude. The segment from the vertex to the circumcenter is then $2 \times 3.2 \text{ cm} = 6.4 \text{ cm}$, which is 2/3 of the total altitude. The total altitude is $3.2 \text{ cm} + 6.4 \text{ cm} = 9.6 \text{ cm}$. This confirms our calculation.

Summary of Calculation

Revision Table: Equilateral Triangle Properties

Additional Information: Geometric Centers

Understanding the different geometric centers of a triangle is important:

• Circumcenter: The intersection of the perpendicular bisectors of the sides. It is the center of the circumcircle (circle passing through vertices).
• Incenter: The intersection of the angle bisectors. It is the center of the incircle (circle tangent to sides).
• Centroid: The intersection of the medians (segments from a vertex to the midpoint of the opposite side). It is the center of mass of the triangle.
• Orthocenter: The intersection of the altitudes (segments from a vertex perpendicular to the opposite side).

In an equilateral triangle, these four points are the same. In other triangles, they are generally distinct, except for isosceles triangles where they are collinear.

The 2:1 ratio property is specific to the centroid dividing the median. Since the centroid and circumcenter are the same in an equilateral triangle, this ratio applies to the position of the circumcenter on the altitude.