Triangle A B C is a right triangle with altitude segment C D.
A diagram shows triangle A B C. Angle A C B is a right angle. Point D is on side A B, opposite the right angle.
Question
Move the options into the spaces to complete the proof of the Pythagorean Theorem for triangle A B C.
Response area with 4 blank spaces
Because segment C D is an altitude of a right triangle, triangles C B D and A C D are similar to triangle A B C. Thus, StartFraction B D over B C EndFraction equals StartFraction B C over B A EndFraction and StartFraction A D over A C EndFraction equals StartFraction A C over A B EndFraction. By cross-multiplication, the equations become
Blank space 11 empty
and
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. These two equations add and simplify to become left-parenthesis A C right-parenthesis squared plus left-parenthesis B C right-parenthesis squared equals
Blank space 13 empty
. By segment addition, B A equals A D plus B D. Thus, by substitution, left-parenthesis A C right-parenthesis squared plus left-parenthesis B C right-parenthesis squared equals
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.
Answer options with 7 options.
Blank space 11: B D/B C
Blank space 12: A D/A C
Blank space 13: A B^2
Blank space 14: (A D + B D)^2
Answer options: 1) B A^2, 2) A C^2, 3) B D^2, 4) B C^2, 5) A C^2 + B C^2, 6) A D^2 + B D^2, 7) A C^2 + B C^2 + A D^2 + B D^2