If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
We can also write this theorem as follows.
Given: & â–³ ABC is a right triangle
& CD is the altitude to the hypotenuse
Prove: & â–³ ACD ~ â–³ ABC, â–³ CBD ~ â–³ ABC
& â–³ ACD ~ â–³ CBD
Notice that ∠1 and ∠2 are right angles because altitude always forms a right angle with the opposite side. Also, from the graph we know that ∠ACB = 90^(∘). Therefore, we can say that ∠1, ∠2, and ∠ACB are congruent since they are all right angles.
∠1 ≅ ∠ACB ≅ ∠2
Now, by the Reflexive Property of Congruence, we have that ∠A ≅ ∠A and also ∠B ≅ ∠B. With these two congruences, as well as the ones written earlier, we can use the Angle-Angle (AA) Similarity to show that pairs of triangles are similar.
We have shown that the two formed triangles are similar to the original triangle. Finally, by the Transitive Property of Similar Triangles, we also have that â–³ ACD ~ â–³ CBD.
â–³ ACD ~ â–³ ABC &and â–³ ABC ~ â–³ CBD
&⇓
â–³ ACD &~ â–³ CBD
