Question

The figure shows right triangle Upper A Upper B Upper C with altitude segment Upper C Upper D, where triangle Upper A Upper C Upper D is similar to triangle Upper C Upper B Upper D is similar to triangle Upper A Upper B Upper C.



Question
Move the options to the spaces to prove the Pythagorean Theorem.

Response area with 5 blank spaces
Statement Reason
triangle Upper A Upper C Upper D is similar to triangle Upper C Upper B Upper D is similar to triangle Upper A Upper B Upper C

Given

StartFraction a over c EndFraction equals
Blank space 27 empty
, StartFraction b over c EndFraction equals
Blank space 28 empty

Corresponding sides of similar triangles are proportional.

a squared equals c x, b squared equals c y

Blank space 29 empty

a squared plus b squared equals c x plus c y

Blank space 30 empty

a squared plus b squared equals c left-parenthesis x plus y right-parenthesis

Distributive property

a squared plus b squared equals c squared

Blank

Answer 1

Blank 27: x/a
Blank 28: y/b
Blank 29 (reason): Add the two equations (Addition Property of Equality)
Blank 30 (statement): x + y = c

So the filled steps:

- a/c = x/a and b/c = y/b (corresponding sides of similar triangles)
- cross-multiply → a^2 = c x and b^2 = c y
- add the equations → a^2 + b^2 = c x + c y
- since x + y = c (AD + DB = AB), rewrite → a^2 + b^2 = c(x + y)
- distributive / substitution → a^2 + b^2 = c^2.