SOLUTION: Given: Right triangle RST with hypotenuse ST and altitude RU Prove: (RS)^2 + (RT)^2 = ST^2

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Question 1079704: Given: Right triangle RST with hypotenuse ST and altitude RU
Prove: (RS)^2 + (RT)^2 = ST^2
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

∡T ≅ ∡T          reflexive property
∡SRT ≅ ∡RUT      both are right angles
ΔSTR ∽ ΔRTU      two angles congruent in each

TU%2F%28RT%29%22%22=%22%22RT%2FST   CPST

TU%2AST%22%22=%22%22RT%5E2    Cross-multiplying

---

∡S ≅ ∡S          reflexive property
∡SRT ≅ ∡SUR      both are right angles
ΔSTR ∽ ΔSRU      two angles congruent in each

AE%2F%28RS%29%22%22=%22%22RS%2FST   CPST

AE%2AST%22%22=%22%22RS%5E2      Cross-multiplying

TU%2AST%2BSU%2AST%22%22=%22%22RT%5E2%2BRS%5E2  Equals added to equals

Factor out ST on the left side:

ST%28TU%2BSU%29%22%22=%22%22RT%5E2%2BRS%5E2

TU+SU = ST                     Whole = sum of parts

ST%28ST%29%22%22=%22%22RT%5E2%2BRS%5E2  Replacing equal by equal

ST%5E2%22%22=%22%22RT%5E2%2BRS%5E2   

Same as what you had to prove.

Edwin

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
For other proofs (including the original Pythagoras' proof) see the lessons
    - The Pythagorean Theorem
    - More proofs of the Pythagorean Theorem
in this site.


Also,  you have this free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lessons are the part if this textbook under the topic
"Right-angled triangles. The Pythagorean theorem. Properties of right-angled triangles".