Question

consider the diagram and the paragraph proof below.
given: right △abc as shown where cd is an altitude of the triangle
prove: ( a^2 + b^2 = c^2 )
diagram of right triangle abc with cd as altitude, labels: a, b, c, d, e, f, c, a, b, h
because △abc and △cbd both have a right angle, and the same angle b is in both triangles, the triangles must be similar by aa. likewise, △abc and △acd both have a right angle, and the same angle a is in both triangles, so they also must be similar by aa. the proportions ( \frac{c}{a} = \frac{a}{f} ) and ( \frac{c}{b} = \frac{b}{e} ) are true
which is the last sentence of the proof?
○ because ( f + e = 1 ), ( a^2 + b^2 = c^2 ).
○ because ( f + e = c ), ( a^2 + b^2 = c^2 ).
○ because ( a^2 + b^2 = c^2 ), ( f + e = c ).
○ because ( a^2 + b^2 = c^2 ), ( f + e = 1 ).

Explanation:

From the diagram, \( e + f = c \) (segment addition on hypotenuse \( AB \)). From similar triangles, we derive \( a^{2}=cf \) and \( b^{2}=ce \). Adding these: \( a^{2}+b^{2}=c(f + e) \). Since \( f + e = c \), substitute to get \( a^{2}+b^{2}=c^{2} \). So the last sentence uses \( f + e = c \) to conclude \( a^{2}+b^{2}=c^{2} \).

Answer:

Because \( f + e = c \), \( a^{2}+b^{2}=c^{2} \)