Two perpendicular medians in a triangle.
Geometry
Level
3
In the figure shown, the two medians in the triangle ABC, AM and CN are perpendicular to each other.
If a,b and c are the sides of such a triangle,
Find
(a^2 + c^2)/b^2
The answer is 5.
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1 solution
Since the point of intersection of the two medians is the centroid O
Use the property of a centroid which divides any median in the ratio 2: 1
and
Set AO = 2m, and CO = 2n,
and Set OM = m and ON = n.
Apply Pythagoras theorems on triangles: AON, COM and AOC to arrive at:
(a^2 + c^2)/b^2 = 5
Note that AN = c/2 and CM = a/2 since AM and AN are medians.