Question 30
Professor Jussara likes to make reduced copies. She starts by drawing the right triangle $A B C$ below: 
Next, Professor Jussara traces a segment $A P_{1}$ so that $A P_{1}$ is perpendicular to $B C$, as shown in the figure below. 
Jussara then states that the triangle $A P_{1} C$ is similar to the triangle $A B C$ (that is, it has equal corresponding angles).
a) Show that teacher Jussara is right!
b) Calculate the length of segment $A P_{1}$.
c) Calculate the ratio between the area of the triangle $A B C$ and the area of the triangle $A P_{1} C$.
d) Professor Jussara repeats the process, now drawing a segment $P_{1} P_{2}$ perpendicular to the side $A C$, as shown in the figure below.
What is the ratio between the areas of the triangles $P_{1} P_{2} C$ and $A P_{1} C$ ?
e) Professor Jussara repeats the process twice more, as shown in the figure below.
What is the area of the triangle $P_{5} P_{6} C$ ?
Next, Professor Jussara traces a segment $A P_{1}$ so that $A P_{1}$ is perpendicular to $B C$, as shown in the figure below. Jussara then states that the triangle $A P_{1} C$ is similar to the triangle $A B C$ (that is, it has equal corresponding angles). a) Show that teacher Jussara is right!
b) Calculate the length of segment $A P_{1}$.
c) Calculate the ratio between the area of the triangle $A B C$ and the area of the triangle $A P_{1} C$.
d) Professor Jussara repeats the process, now drawing a segment $P_{1} P_{2}$ perpendicular to the side $A C$, as shown in the figure below.
What is the ratio between the areas of the triangles $P_{1} P_{2} C$ and $A P_{1} C$ ? e) Professor Jussara repeats the process twice more, as shown in the figure below.
What is the area of the triangle $P_{5} P_{6} C$ ?