Medians of a Triangle

Mathematics — Second Preparatory

Worked examples

Comparison aspect	Main idea	Keywords
Example 1	In triangle ABC the median from A is AM = 12 cm and the centroid G lies on it. Find AG and GM	trianglemediancentroidlies
Solution	AG = (2/3) × 12 = 8 cm and GM = 12 − 8 = 4 cm. Check: 8 : 4 = 2 : 1	Check
Example 2	On the coordinate plane the vertices of a triangle are A(0 B(6 0) and C(0 6). Find the midpoint of BC and the centroid then check the 2 : 1 property for the median from A	coordinateplaneverticestriangle
Solution	The midpoint of BC is M(3, 3). The centroid is G = ((0 + 6 + 0) ÷ 3, (0 + 0 + 6) ÷ 3) = (2, 2). The length AM = √(3² + 3²) = √18, and AG = √(2² + 2²) = √8. Since √8 = (2/3) × √18, the property holds.	midpointcentroidlengthSince
Example 3	Find the coordinates of the centroid of the triangle with vertices A(1 B(5 2) and C(3	Findcoordinatescentroidtriangle

Key points

SolutionG = ((1 + 5 + 3) ÷ 3, (2 + 2 + 8) ÷ 3) = (3, 4).

Example 4The area of triangle ABC is 36 cm², and AM is a median. Find the area of triangle ABM.

SolutionThe median divides the triangle into two triangles of equal area, so the area of ABM = 36 ÷ 2 = 18 cm².

Medians of a Triangle · 4 / 6