Question

Given: ∆BCA is a right triangle.
Prove: a2 + b2 = c2
Statement
Justification
Draw an altitude from point C to Line Segment AB

Let Line Segment BC= a
Line Segment CA= b
Line Segment AB= c
Line Segment CD= h
Line Segment DB= x
Line Segment AD= y

y + x = c

c over a is equal to a over y; c over b is equal to b over x

a2 = cy; b2 = cx

a2 + b2 = cy + b2

a2 + b2 = cy + cx

a2 + b2 = c(y + x)

a2 + b2 = c(c)

a2 + b2 = c2



Which is NOT a justification for the proof?
Addition Property of Equality

Pythagorean Theorem

Pieces of Right Triangles Similarity Theorem

Cross Product Property

Answer 1

Notice we have a^2 = cy

We then add b^2 to both sides to get

a^2 + b^2 = cy + b^2

We're using the Addition Property of Equality to justify this move, so we can rule this out.

So we can eliminate choice D as a possible answer.

Answer 2

We then go from

a^2 + b^2 = cy + b^2

to

a^2 + b^2 = cy + cx


we can do this because b^2 = cx, so this is using the Substitution property (ie replace one thing with something that it's equal to)


So choice C is out

Answer 3

And finally, when we cut triangle BCA by drawing in that altitude, we're creating 2 similar smaller triangles. We're then exploiting the fact that they're similar to form these various ratios.

This is all justified by the "Pieces of Right Triangles Similarity Theorem" which says that if you draw an altitude from the 90 degree angle to the opposite side, then you will have created 2 similar smaller triangles (that are similar to the largest one)


So choice A is out

Answer 4

So it looks like it's choice B

Answer 5

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