The Centroid and Its Role in Triangles

Step-by-step Guide: The Centroid of a Triangle

1. Definition of a Centroid:
The centroid of a triangle is the point of intersection of its three medians. A median is a segment joining a vertex of the triangle to the midpoint of the opposite side. Given that every triangle has three vertices and three sides, it will have three medians, and they all intersect at the centroid.

2. Properties of the Centroid:

• Division of Medians: The centroid divides each median in a \(2:1\) ratio, with the segment from the vertex to the centroid being twice as long as the segment from the centroid to the midpoint of the side.
• Center of Gravity: The centroid acts as the center of gravity or the balance point of a triangle. If a triangle were to be made out of a uniform material, it would balance perfectly on its centroid.
• Area Distribution: When the medians intersect at the centroid, they divide the triangle into six smaller triangles. Interestingly, all these triangles have equal areas.

Examples

Example 1:
Given a triangle \(ABC\), if the median \(AD\) (from vertex \(A\) to the midpoint \(D\) of side \(BC\)) has a length of \(20 \text{ cm} \), how far is the centroid (let’s call it \(G\)) from vertex \(A\) and from point \(D\)?

Solution:
Using the \(2:1\) division ratio of the centroid:
Distance from \(A\) to \(G = \frac{2}{3} \times 20 \text{ cm} = 13.33 \text{ cm} \)
Distance from \(G\) to \(D = \frac{1}{3} \times 20 \text{ cm} = 6.67 \text{ cm} \)

Example 2:
In triangle \(PQR\), if the distance from vertex \(P\) to the centroid (let’s call it \(M\)) along the median is \(12 \text{ cm} \), what is the distance from \(M\) to the midpoint of side \(QR\)?

Solution:
Given the \(2:1\) ratio of the centroid:
Distance from \(M\) to midpoint of \(QR = \frac{1}{2} \times 12 \text{ cm} = 6 \text{ cm} \)

Practice Questions:

1. In a triangle, if the median’s length is \(24\text{ cm} \), how far is the centroid from the midpoint of the side the median is drawn to?
2. If the distance from a vertex to the centroid along a median in a triangle is \(15\text{ cm} \), what is the total length of this median?

Answers:

1. \( \frac{1}{3} \times 24 \text{ cm} = 8 \text{ cm} \)
2. Given the \(2:1\) ratio, the total median length \(= 15 \text{ cm} \times \frac{3}{2} = 22.5 \text{ cm} \)