Question

see the proof of ( ab = 2am ), and ( am = \frac{1}{2}ab ) from example 4 below. how would the proof be different if you were proving that ( ab = 2mb ) and that ( mb = \frac{1}{2}ab ) instead?
given ( m ) is the midpoint of ( overline{ab} ).
prove ( ab = 2am, am = \frac{1}{2}ab )
diagram of segment ( ab ) with ( m ) between ( a ) and ( b )

statements	reasons
2. ( overline{am} cong overline{mb} )	2. definition of midpoint
3. ( am = mb )	3. definition of congruent segments
4. ( am + mb = ab )	4. segment addition postulate
5. ( am + am = ab )	5. substitution property of equality
6. ( 2am = ab )	6. distributive property
7. ( am = \frac{1}{2}ab )	7. division property of equality

in steps 5, 6, and 7, mb dropdown would be replaced by am dropdown.
previous, 1, 2, am, ab, mb, next buttons

Explanation:

Step1: Analyze the original proof

In the original proof, we used \( AM = MB \) (from midpoint) and substituted \( MB \) with \( AM \) in \( AM + MB = AB \) to get \( AM + AM = AB \).

Step2: Adjust for proving \( AB = 2MB \) and \( MB=\frac{1}{2}AB \)

• Step 5: Instead of \( AM + AM = AB \), we substitute \( AM \) with \( MB \) (since \( AM = MB \)) in \( AM + MB = AB \), so we get \( MB + MB = AB \).
• Step 6: Simplify \( MB + MB = AB \) using the distributive property: \( 2MB = AB \).
• Step 7: Use the division property of equality on \( 2MB = AB \) to get \( MB=\frac{1}{2}AB \).

So the key difference is that in steps 5, 6, and 7, we substitute with \( MB \) instead of \( AM \) (or replace \( AM \) with \( MB \) in the substitution step).

Answer:

In Steps 5, 6, and 7, \( AM \) would be replaced by \( MB \). Specifically, Step 5 becomes \( MB + MB = AB \), Step 6 becomes \( 2MB = AB \), and Step 7 becomes \( MB=\frac{1}{2}AB \), using the same properties (substitution, distributive, division) but with \( MB \) instead of \( AM \).