Similar Triangles - ratios of parts

In two similar triangles, their perimeters and corresponding sides, medians and altitudes will all be in the same ratio.

Try this The two triangles below are similar. Drag any orange dot at P,Q,R. Note the ratio of the two corresponding sides and the ratio of the medians.

In two similar triangles:

  • The perimeters of the two triangles are in the same ratio as the sides.
  • The corresponding sides, medians and altitudes will all be in this same ratio.

This is illustrated by the two similar triangles in the figure above. Here are shown one of the medians of each triangle. As you resize the triangle PQR, you can see that the ratio of the sides is always equal to the ratio of the medians. In the same way, the perimeters will be in the same ratio and the altitudes will also be in the same ratio.

There are ten items altogether that are in the same ratio (perimeter, three medians, three altitudes, three sides). To avoid cluttering up the diagram above, the ratios of just one side and one median are shown, but the idea applies to all ten items.

Remember that a triangle has three sides, three altitudes and three medians. Be sure to compare the corresponding part in each triangle. For example in the figure below the two altitudes are corresponding altitudes because they are drawn form the same vertex in each triangle, and so the ratio of their lengths is valid.

Two corresponding altitudes of similar triangles

Be especially vigilant where one triangle is rotated and / or mirror-image of the other. To see this, in the figure at the top drag the point P down below Q. The triangle will be rotated 180° but the triangles are still similar and the ratios still hold.

See also Similar Triangles - ratios of areas

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Other similarity topics

Similar Polygons