Does the similar figures theorem apply also to altitudes and medians?
1 Answer
Yes.
Explanation:
A similar figure is a figure that is geometrically the same.
In the case of triangles, two triangles are similar if their sides are proportionate.
I assume we are talking about the theorem of scaling.
If we have a triangle and we scale by a factor
Since a triangle is uniquely defined by the length of its sides, all angles will remain the same and consequently the altitude will increase by
The medians will still be in the same relative position, by definition of a median. The length from a vertex to the opposite side will increase by the factor
The easiest way to see this is:
All linear measurements i.e. length will increase by
As an example:
Given an equilateral triangle with sides 4.
Scaled by a factor
Altitude of triangle:
After scaling by
Altitude of scaled triangle:
Notice this is:
Notice area:
Area of original triangle:
Area of scaled triangle:
As you can see the area is quadratic, so we used
Don't know whether this helps you.