Does the similar figures theorem apply also to altitudes and medians?

1 Answer
Apr 26, 2018

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 a/b, all sides of the scaled triangle will be increased by a/b.

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 a/b

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 a/b. ( this is a linear measurement )

The easiest way to see this is:

All linear measurements i.e. length will increase by a/b, all quadratic measurements will increase by (a/b)^2 and all cubic measurements will increase by (a/b)^3. Angular measurement remain unchanged by scaling, these are non linear, non quadratic and non cubic.

As an example:

Given an equilateral triangle with sides 4.

Scaled by a factor 1/2

Altitude of triangle:

sqrt(4^2-2^2)=sqrt(12)=2sqrt(3)

After scaling by 1/2:

"sides"=2

Altitude of scaled triangle:

sqrt(2^2-1^2)=sqrt(3)

Notice this is:

1/2xx2sqrt(3)=sqrt(3)

Notice area:

Area of original triangle:

1/2(4)xx2sqrt(3)=4sqrt(3)

Area of scaled triangle:

1/2(2)xxsqrt(3)=sqrt(3)

(1/2)^2xx4sqrt(3)=1/4xx4sqrt(3)=sqrt(3)

As you can see the area is quadratic, so we used (a/b)^2

Don't know whether this helps you.