Special Segments in Similar Triangles

In two similar triangles the corresponding sides are all in the same ratio k. The special segments inside a triangle — its altitudes, angle bisectors, and medians — obey exactly the same rule.

Theorem The golden rule

In two similar triangles, every pair of corresponding segments — sides, altitudes, angle bisectors, and medians — is in the same ratio k.

Altitude — the perpendicular from a vertex to the opposite side.

Angle bisector — splits an angle into two equal halves.

Median — joins a vertex to the midpoint of the opposite side.

The two altitudes follow the same ratio as the sides: 60 ÷ 30 = 2, so k = 2.

Example Finding a segment

Two triangles are similar with ratio $k = 3$ . An altitude of the larger triangle is 9. How long is the corresponding altitude of the smaller one?

corresponding altitude

= \dfrac{9}{k} = \dfrac{9}{3}

⟹ the corresponding altitude is 3.

Example Finding the ratio

Two triangles are similar. A median of the first is 12 and the corresponding median of the second is 4. Find the similarity ratio.

k = \dfrac{12}{4}

⟹ k = 3.

Summary

The golden rule: in similar triangles, all corresponding segments share the same ratio k.
Every triangle has 3 altitudes, 3 angle bisectors, and 3 medians — each corresponding pair is in ratio k.
The three medians meet at the centroid, which divides each median in a 2 : 1 ratio.
Careful: the ratio of the two areas is k², not k.