Comparison and differences between altitudes and medians in a triangle
| Aspect | Median | Altitude |
|---|---|---|
| Definition | A line segment from a vertex to the midpoint of the | A line segment from a vertex to the opposite side (or its extension) and perpendicular to it. |
| Purpose | Divides the opposite side into two | Represents the shortest distance from a vertex to the opposite side. |
| Intersection | The point where it intersects the | The point where it intersects the opposite side is called the foot of the altitude. |
| Properties | In any triangle, the medians intersect at the | In any triangle, the altitudes intersect at the point called orthocentre, which may lie inside or outside the |
| Dependence on Triangle Type | In an isosceles or equilateral triangle, a median can also be an altitude. | In a right-angled triangle, the two legs can serve as altitudes. In an |
| In Equilateral Triangle | All medians are equal in length and also serve as altitudes. | All altitudes are |
1. Take several cut-outs of
a
(i) an equilateral triangle
equilateral triangle
Find their altitudes and medians. Do you find anything special about them? Discuss it with your friends.
Solution:
equilateral triangle
In an equilateral triangle, all medians, altitudes, angle bisectors, and perpendicular bisectors coincide. This is a unique property of equilateral triangles altitude and median are
b
(ii) an isosceles triangle and
isosceles triangle
Find their altitudes and medians. Do you find anything special about them? Discuss it with your friends.
Solution:
isosceles triangle
In an isosceles triangle, the median from the vertex angle to the base is also an altitude and an angle bisector, but this is not true for the other medians altitude and median are
c
(iii) a scalene triangle.
scalene triangle
Find their altitudes and medians. Do you find anything special about them? Discuss it with your friends.
Solution:
scalene triangle
In a scalene triangle, the altitudes and medians are all distinct and do not necessarily coincide altitude and median are