Shortest Proofs of the Pythagorean Theorem

Set 1.4
David Cohen
Question 32

See figure below first attachment. I would like all four parts solved as additional math notes for self-study.

 

a)
the following is sufficient to say that two right triangles are similar:
If all three angles of a triangle are congruent but the sides are not, then one of the triangles is a scaled up version of the other. When this happens the proportions between the sides still remains unchanged which is the criteria for similarity.

if <, CAD = <DCB, and <ADC=<BDC=90=> then <ACD=<DCB; so, all three angles of a triangle are congruent =>two right triangles BCD and BAC are similar and corresponding sides are proportional

b)
corresponding sides in triangles BCD and BAC are:
a and c (hypothenuses)
y and a (legs)

then
a/y=c/a ...........cross multiply to solve for a
a*a=c*y
a^2=cy

c) show that triangles ACD and ABC are similar
<ADC=<BCA=90
<, CAD = <DCB=>then <ACD=<DCB; so, all three angles of a triangle are congruent =>two right triangles ACD and ABC are similar and corresponding sides are proportional

corresponding sides in triangles ACD and ABC are:
b and c (hypothenuses)
c-y and b (legs)

b/c=(c-y)/b
b*b=c(c-y)
b^2=c^2-cy

d)
from part b) we have a^2=cy.....eq.1
from part c) we have b^2=c^2-cy........eq.2

b^2=c^2-cy......eq.2..........substitute cy from eq.1

b^2=c^2-a^2.............move a^2 to the left
a^2+ b^2=c^2

 

Wow! I'm impressed. You are a true mathematician.

 

Can you please check out my thread Triangle ABC?
Your help is amazing. I thank you for taking this journey with me.