To prove the Pythagorean Theorem in a right triangle \( \Delta ABC \) where \( C \) is the right angle, we can start by establishing similar triangles within the figure. The first step that aligns with this method is:
C. Use the corollary 'Geometric Mean in Similar Right Triangles: Legs.'
This involves demonstrating relationships between the right triangle and the segments created by dropping a perpendicular from point \( C \) to \( AB \). By utilizing similar triangles, we can derive the necessary relationships between the lengths of the sides \( a \), \( b \), and \( c \).
To outline further steps after this initial decision, you would typically proceed as follows:
- Drop a perpendicular from \( C \) to line segment \( AB \), calling the point of intersection \( D \).
- Show that triangles \( \Delta ACD \) and \( \Delta BCD \) are similar to triangle \( \Delta ABC \) and to each other.
- Establish the relationships \( AD^2 = AC^2 - DC^2 \) and \( BD^2 = BC^2 - DC^2 \) based on the similarities of triangles.
- Combine these relations to come to the conclusion that \( a^2 + b^2 = c^2 \).
By relying on the properties of similar triangles and their proportions, we can ultimately arrive at the proof for the Pythagorean Theorem.