Prove the Pythagorean Theorem.

Question

Given that Upper Delta ABC is a right​ triangle, prove asquaredplusbsquaredequalscsquared.
A
B
C
a
b
c
q
r
D
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Part 1
What is the first step to be​ performed?
A.
Use the corollary​ 'Geometric Mean in Similar Right​ Triangles: Hypotenuse.'
B.
Use the Angle Bisector Theorem.
C.
Use the corollary​ 'Geometric Mean in Similar Right​ Triangles: Legs.'
D.
Use the Hinges Theorem.

Answer 1

To prove the Pythagorean Theorem in a right triangle $\mathrm{\Delta }ABC$ 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:

1. Drop a perpendicular from $C$ to line segment $AB$, calling the point of intersection $D$.
2. Show that triangles $\mathrm{\Delta }ACD$ and $\mathrm{\Delta }BCD$ are similar to triangle $\mathrm{\Delta }ABC$ and to each other.
3. Establish the relationships $A{D}^2=A{C}^2-D{C}^2$ and $B{D}^2=B{C}^2-D{C}^2$ based on the similarities of triangles.
4. 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.