We worked through this worksheet:

Geometric Mean (back practice problems.)

On the front (not back practice problems), first separate the given diagram into the three triangles, ACD, DCB, and ABC.

Each triangle is a right triangle. Keep all points and sides labeled with the letters given on the original diagram, as well as all right angle marks.

All three triangles already have one angle in common. If there were a second, they would all be similar by AA.

If you drew your triangles in order from smallest to largest, you should notice that the 1st and 3rd triangles both have angle A as one of their angles. We can say that <A is congruent to <A by the reflexive property.

The same idea applies to the second and third triangle. They have angle B in common, so <B is congruent to <B by the reflexive property.

We can then use the Third Angles Theorem to show the unmarked angles remaining on the first and second triangles having other angles in common.

So all three triangles are similar. We create a similarity statement in three parts connecting all three triangles. Now we use CSSTP (Corresponding Sides of Similar Triangles are Proportional) to create proportions between various sides.