Special Points In Triangles Revision Notes for Leaving Cert Mathematics

Special Points In Triangles

Overview

Triangles have special points where specific lines or segments intersect. These points are crucial in geometry due to their unique properties. Two such concepts are the centroid, where the medians of a triangle meet, and the midpoint line property, which relates the midpoints of two sides to the third side.

Concurrent Medians

Statement: The medians of a triangle intersect at a single point called the centroid. This point divides each median in the ratio $2:1$ , with the larger segment being between the vertex and the centroid.

Why It Works: The centroid is the balance point (centre of gravity) of the triangle. The $2:1$ division arises from the geometric relationship between the triangle's vertices and the centroid.

The Line Joining Midpoints (Midpoint Theorem)

Statement: The line joining the midpoints of two sides of a triangle is:

Parallel to the third side.
Half the length of the third side. Why It Works:

The line connecting the midpoints forms a smaller triangle similar to the original triangle, preserving parallelism and proportionality.

Worked Examples

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Example 1: Concurrent Medians

Problem: In $\triangle ABC$ , medians $AD$ , $BE$ , and $CF$ intersect at $G$ .

If $AG = 6$ , find the lengths of $GD$ and $AD$ .

Solution:

Step 1: The centroid divides each median in the ratio $2:1$

Step 2: Since $AG = 6$ , $GD$ ** is one-third of** $AD$

$GD = \frac{1}{2} \times AG = 3$

Step 3: The total length of $AD$

$AD = AG + GD = 6 + 3 = 9$

Answer: $GD = 3$ , and $AD = 9$

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Example 2: The Line Joining Midpoints

Problem: In $\triangle ABC$ , $D$ and $E$ are the midpoints of $AB$ and $AC$ , respectively.

If $BC = 12$ , prove that $DE \parallel BC$ and find $DE$ .

Solution:

Step 1: By the Midpoint Theorem, $DE \parallel BC$

Step 2: The length of $DE$ ** is half the length of** $BC$

$DE = \frac{1}{2} \times BC = \frac{1}{2} \times 12 = 6$

Answer: $DE \parallel BC$ , and $DE = 6$

Summary

Concurrent Medians (Centroid): The medians of a triangle intersect at the centroid, dividing each median in the ratio $2:1$
Midpoint Theorem: The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
These special points and relationships simplify triangle constructions and problem-solving in geometry.

Special Points In Triangles (Leaving Cert Mathematics): Revision Notes